问:
Actually Integer Metagolf

Background

Actually (the successor to Seriously) is a stack-based imperative golfing language I created in November of 2015. Like many other golfing languages, it has one-byte commands that perform different functions based on the contents of the stack. One of its specialties is mathematics - it has a wide variety of mathematics-based commands. However, in order to perform math operations, you have to put (one or more) numbers on the stack. Pushing a specific value in as few bytes as possible is tricky, because there are many different options. In this challenge, you will be doing exactly that: representing numbers (specifically, integers) in Actually in as few bytes as possible.

The Challenge

Given an integer N as input, output valid Actually code that results in N being pushed to the stack.

The input will be within the range of a 32-bit signed two's complement integer (i.e. the integers in the inclusive range [-2147483648, 2147483647]).

The result must be an integer (not a float, string, list, or function), and must be on top of the stack.

You may not make any assumptions about the contents of the stack (like whether or not it is empty). Any existing values on the stack must not be modified or rearranged.

The latest commit of Actually at the time I am writing this challenge is to be used. If I make bug fixes or performance enhancements (or any other minor change that does not remove or change functionality of the allowed commands), I will update this version.

Your solution must do at least as well as the trivial solution (prepend : to the input to make a numeric literal).

Your score will be the sum of the lengths of the trivial solutions minus the sum of the lengths of the outputs for the selection of 1000 32-bit signed two's complement integers used for scoring, which can be found below. I reserve the right to change the scoring integers at any time, if there is a need (such as optimizing for the test cases or the test cases not being thorough enough).

Solutions must output a valid answer within 30 seconds for each input. Timing will be done on a standard free Cloud9 workspace.

Commands

For simplicity, only 141 out of the (currently) 208 commands may be used, and many overloads of those 141 that are unrelated to number-crunching have been removed. Here is the list of allowed commands (format is <hex code> (<symbol>): <descriptions of functions based on stack values and types>:

```<code>0B (♂): take the next command and map it over the top of the stack (for example, ♂A is equivalent to `A`M)
1F (▼): pop a,b: push b//gcd(a,b),a//gcd(a,b); pop [a]: push [x//gcd([a]) for x in [a]]
20 ( ): push the # of elements on the stack (push len(stack))
21 (!): pop a: push a! (factorial(a))
22 ("): string literal, reads until next " and pushes value onto stack. An implied " is present at EOF if needed.
23 (#): pop a: push list(a)
25 (%): pop a,b: push a%b
26 (&): pop a,b: push a & b (bitwise AND)
27 ('): pushes next character onto stack as character literal (length-1 string)
28 ((): rotates stack right by 1
29 ()): rotates stack left by 1
2A (*): pop a,b: push a*b; pop a,[b] or [b],a: apply a* to each element in the array; pop [a],[b]: push dot product of [a] and [b] (sum([a[i]*b[i] for i in len(a)])) (shorter is padded with 0s)
2B (+): pop a,b: push a+b; pop [a],[b]: push [a][b] (append [b] to [a]); pop a,[b] or [b],a: apply a+ to each element in the array
2D (-): pop a,b: push a-b; pop [a],[b]: push [a]-[b] (all elements of [a] not in [b])
2F (/): pop a,b: push a/b (float division); pop [a]: rotate [a] right by 1, push [a]
30 (0): push 0
31 (1): push 1
32 (2): push 2
33 (3): push 3
34 (4): push 4
35 (5): push 5
36 (6): push 6
37 (7): push 7
38 (8): push 8
39 (9): push 9
3A (:): numeric literal delimiter: pushes the longest string of characters in '0123456789+-.ij' as a numeric
3B (;): pop a: push a,a (duplicates top element)
3C (<): pop a,b: push 1 if a<b else 0
3D (=): pop a,b: push 1 if a==b else 0
3E (>): pop a,b: push 1 if a>b else 0
40 (@): pop a,b: push a,b (rotate top 2 elements)
41 (A): pop a: push abs(a)
43 (C): pop a: push cos(a)
44 (D): pop a: push a-1; pop [a]: push stddev([a])
45 (E): pop a: push erf(a); pop [a],b: push [a][b] (bth item in [a])
46 (F): pop a: push Fib(a) (Fib(0)=0, Fib(1)=Fib(2)=1); pop [a]: push a[0]
48 (H): pop [a],b: push [a][:b]
49 (I): pop a,b,c: push b if a is truthy, else push c
4B (K): pop a: push ceil(a)
4C (L): pop a: push floor(a)
4D (M): pop f,[a], execute f for each element in [a], using the element as a temporary stack, push [a] (similar to map(f,[a])); pop [a]: push max([a])
4E (N): pop [a]: push a[-1]
50 (P): pop a: push the a-th prime (zero-indexed)
52 (R): pop f,[a]: call f, using [a] as a temporary stack, push [a] (similar to reduce(f,[a])); pop [a]: push [a][::-1] (reverse); pop a: push [1,2,...,a] (range(1,a+1))
53 (S): pop a: push sin(a); pop [a]: push sorted(a)
54 (T): pop a: push tan(a); pop [a],b,c: set [a][b] to c, push [a]
55 (U): pop [a],[b]: push union of [a] and [b]
57 (W): loop delimiter: peek top of stack, repeat code in loop while a evaluates to true
58 (X): pop a: discard
59 (Y): pop a: push !bool(a) (logical negate, opposite of b)
5A (Z): pop [a],[b]: push zip([a],[b]); pop a, zip the next a lists
5B ([): begin list literal, values are delimited by commas (,)
5C (\): pop a,b: push a/b (integer division); pop [a]: rotate [a] left by 1, push [a]
5D (]): end list literal
5E (^): pop a,b: push a XOR b
5F (_): pop a: push ln(a)
60 (`): function literal delimiter, pushes function whose body contains all of the commands until the next `. An implied ` is present at EOF if needed.
61 (a): invert the stack ([a,b,c,d] -> [d,c,b,a])
62 (b): pop a: push 0 if a==0 else 1; pop "a" or [a]: push 0 if len(a)==0 else 1; pop f: push 0 if len(f)==0 else 1
63 (c): pop a: push character at ordinal a%256; pop [a],b: push [a].count(b)
64 (d): pop [a]: dequeue b from [a], push [a],b
65 (e): pop a: push exp(a)
66 (f): pop a: push the Fibonacci index of a if a is a Fibonacci number, else -1
67 (g): pop a,b: push gcd(a,b); pop [a]: push gcd([a])
68 (h): pop a,b: push sqrt(a*a+b*b) (Euclidean norm)
69 (i): pop [a]: push each element from [a], starting from end (flatten)
6B (k): pop all elements from stack, convert to list (in the order they were on the stack, starting from the top), and push
6C (l): pop [a]: push len([a])
6D (m): pop a: push int(a),frac(a) (modf(a)); pop [a]: push min([a])
6E (n): pop a,b: push a b times
6F (o): pop [a],b: push b to [a], push [a]
70 (p): pop a: push 1 if a is prime else 0; pop [a]: pop b from [a], push [a],b
71 (q): pop [a],b: enqueue b in [a], push [a]
72 (r): pop a: push [0,1,...,a-1] (range(0,a))
75 (u): pop a: push a+1
77 (w): pop a: push the full positive prime factorization of |a| (18 -> [[2,1],[3,2]], -5 -> [[5,1]])
78 (x): pop a,b: push [a,b) (range(a,b)); pop [a]: push range(*a)
79 (y): pop a: push the positive prime factors of |a| (18 -> [2,3], -5 -> [5])
7B ({): pop a: rotate stack right a times
7C (|): pop a,b: push a | b (bitwise OR)
7D (}): pop a: rotate stack left a times
7E (~): pop a: push ~a (unary bitwise negate)
80 (Ç): pop a,b: push a+bi; pop [a]: pop pairs of real numerics b,c from [a] and push b+ci (appending 0 to [a] if len([a]) is odd)
83 (â): pop a: push asin(a)
84 (ä): pop a: push acos(a)
85 (à): pop a: push atan(a)
86 (å): pop a,b: push atan2(a,b)
87 (ç): pop a: push asinh(a)
88 (ê): pop a: push acosh(a)
89 (ë): pop a: push atanh(a)
8B (ï): push i, the imaginary unit (sqrt(-1) or 0+1i)
8C (î): pop a, push 0+ai; pop [a], push [a] with every element multiplied by i
8D (ì): pop a: push 1/a
8E (Ä): pop a: push sinh(a)
8F (Å): pop a: push cosh(a)
90 (É): pop a: push tanh(a)
91 (æ): pop [a]: push mean([a])
9A (Ü): pop [a]: push mode([a])
9B (¢): pop a,b: push abs(a)*sgn(b)
9D (¥): pop [a],[b]: push the result of pairwise addition of [a] and [b], padding the shorter with 0s
9E (₧): pop z: push phase(z)
A0 (á): pop z: push the complex conjugate of z
A1 (í): pop [a],b: push [a].index(b) (0-based, -1 if not found)
A7 (º): pop a: push degrees(a)
A8 (¿): pop a,b: push int(a,b) (interpret a as a base-b int)
A9 (⌐): pop a: push a+2
AA (¬): pop a: push a-2
AB (½): pop a: push a/2 (float division)
AC (¼): pop a: push a/4 (float division)
AD (¡): pop a,b: push a string representing a in base b
B0 (░): pop [a],[b]: push [[b][i] if [a][i] for i in len(b)], pads [a] with 0s if necessary
B1 (▒): pop a: push totient(a) (# of integers <= a that are coprime with a)
B2 (▓): pop a: push pi(a) (# of primes <= a)
B3 (│): duplicate stack ([a,b,c] => [a,b,c,a,b,c])
B4 (┤): pop a,b: push 1 if a and b are coprime else 0
B9 (╣): pop a: push the ath row in Pascal's triangle
C5 (┼): duplicate each element on stack ([a,b,c] => [a,a,b,b,c,c])
C6 (╞): pop a: make a total copies of each element on stack (3 [a,b,c] -> [a,a,a,b,b,b,c,c,c])
C7 (╟): pop a: pop a elements and push a list containing those elements in their original order
CB (╦): push pi
CC (╠): push e
D1 (╤): pop a: push 10**a
D2 (╥): pop a: push log(a) (log base 10)
D3 (╙): pop a: push 2**a
D4 (╘): pop a: push lg(a) (log base 2)
D5 (╒): push ln(2)
DB (█): pop a,b: push C(a,b) (aCb)
DC (▄): pop a,b: push P(a,b) (aPb)
E2 (Γ): pop a: push Gamma(a)
E3 (π): pop [a]: push product([a])
E4 (Σ): pop [a]: push sum([a])
E7 (τ): pop a: push 2*a
ED (φ): push phi (golden ratio)
F1 (±): pop a: push -a (unary negate)
F2 (≥): pop a,b: push a>=b
F3 (≤): pop a,b: push a<=b
F7 (≈): pop a: push int(a)
F8 (°): pop a: push radians(a)
FB (√): pop a: push sqrt(a)
FC (ⁿ): pop a,b: push pow(a,b)
FD (²): pop a: push a*a

[a] refers to an iterable (list, string, etc.), "a" refers to a string, f refers to a function, and a (and b, and c, and so on) refers to a numeric (or any data type, if the command is type-agnostic).
</code>```

It is likely that many of these commands will not be used, but I have included them just in case they are useful.

Scoring

Here are the integers that will be used for scoring:

```<code>-2124910654
-2117700574
-2098186988
-2095671996
-2083075613
-2058271687
-2052250777
-2024215903
-2019485642
-2016095616
-2009838326
-2009173317
-2007673992
-2000014444
-1999825668
-1992515610
-1989566707
-1975506037
-1955473208
-1950731112
-1949886423
-1920624450
-1918465596
-1916287469
-1905036556
-1903956118
-1888944417
-1865221863
-1856600057
-1842797147
-1835637734
-1812631357
-1805740096
-1798015647
-1726688233
-1723609647
-1713776890
-1700307138
-1687644479
-1645515069
-1617635994
-1610444000
-1579053372
-1556891649
-1549652116
-1537732956
-1535180388
-1527162056
-1524851611
-1524658412
-1523244369
-1521379172
-1520191198
-1519035741
-1516978241
-1508892332
-1489938422
-1482102944
-1481823232
-1470621147
-1469145091
-1446844485
-1441790509
-1437843276
-1435359182
-1434186947
-1429816311
-1429393781
-1419752032
-1400387846
-1385152926
-1372620863
-1369257355
-1361933674
-1360480816
-1334846204
-1323741718
-1323660173
-1312800992
-1308824840
-1304658551
-1287829999
-1283767920
-1276288210
-1264275838
-1263965596
-1262866901
-1255421887
-1251428680
-1244825786
-1243200329
-1235305532
-1233977691
-1220537074
-1214100716
-1199414474
-1190725823
-1190401800
-1178717689
-1172378149
-1147869245
-1142875190
-1138538768
-1137864183
-1124917489
-1102987222
-1095920186
-1083001017
-1080902655
-1047122002
-1031842676
-1029877334
-1020849489
-1001684838
-998419619
-990915088
-985235989
-982515261
-979074894
-974195629
-973832940
-965324937
-944246431
-938387588
-933873331
-932692878
-928039285
-927947459
-914008773
-907981688
-906376330
-903502449
-898112547
-887444438
-862658502
-843628573
-822463032
-786051095
-776932426
-776033951
-752042328
-746532472
-743149468
-740225710
-734414418
-725435852
-708101516
-699674783
-694869277
-693246525
-690571518
-689249770
-688581912
-686864294
-681445866
-647992869
-641101583
-631409299
-624686189
-613079884
-593711206
-591688546
-591331185
-574790069
-573024823
-565387051
-565137163
-556338668
-556291492
-541411509
-538932064
-500479857
-482419890
-468050561
-424532545
-420534171
-408741873
-406973874
-387664799
-382084509
-367095703
-352332569
-352195997
-346430007
-324596389
-320119776
-306594578
-305952425
-283718911
-267302378
-243302738
-242955859
-232180029
-225394407
-217418127
-212286453
-208344820
-191300139
-186177744
-175765723
-161763935
-157025501
-140389149
-132298608
-126175769
-70566352
-68748576
-53985905
-52674668
-50228620
-39678495
-19825663
-8349922
-8186722
-8125700
-8073135
-8043230
-7994382
-7874433
-7863624
-7784916
-7782477
-7696343
-7607278
-7531250
-7388060
-7368780
-7367625
-7353084
-7334489
-7281141
-7267149
-7140057
-7119066
-7010389
-6992089
-6975258
-6863360
-6784772
-6741079
-6585985
-6550401
-6520011
-6495144
-6459702
-6294273
-6178342
-6169344
-6139663
-6090928
-6022637
-5992707
-5971334
-5925304
-5880940
-5873707
-5807953
-5703992
-5692895
-5535131
-5488812
-5468330
-5404148
-5290247
-5274221
-5264144
-5234715
-5224048
-5179837
-5084014
-5043990
-5028537
-5011679
-4998333
-4922901
-4880159
-4874060
-4787390
-4749096
-4736217
-4502308
-4480611
-4424319
-4363262
-4349743
-4290050
-4240069
-4239657
-4174072
-4093051
-4045363
-4037689
-4033352
-3839265
-3766343
-3716899
-3680075
-3679053
-3581776
-3539227
-3461158
-3282526
-3205947
-3183427
-3169708
-3166430
-3089822
-3061531
-2947574
-2930733
-2919246
-2872882
-2830770
-2739228
-2713826
-2634018
-2613990
-2525529
-2439507
-2432921
-2376201
-2335005
-2307524
-2265548
-2176176
-2123133
-2108773
-2105934
-2075032
-2073940
-2045837
-2045648
-1978182
-1945769
-1935486
-1881608
-1654650
-1602520
-1506746
-1505294
-1475309
-1457605
-1327259
-1312217
-1178723
-1027439
-880781
-833776
-666675
-643098
-593446
-468772
-450369
-443225
-418164
-402004
-319964
-307400
-279414
-190199
-176644
-66862
-32745
-32686
-32352
-32261
-32035
-31928
-31414
-31308
-30925
-30411
-29503
-29182
-28573
-28500
-28093
-27743
-27716
-27351
-27201
-26834
-25946
-25019
-24469
-24341
-24292
-24151
-23732
-22769
-22242
-22002
-20863
-20762
-20644
-20189
-20009
-19142
-19036
-18980
-18616
-18196
-18123
-17942
-17915
-17601
-17494
-16669
-16417
-16317
-15051
-14796
-14742
-14600
-14443
-14159
-14046
-13860
-13804
-13745
-13634
-13498
-13497
-12688
-12471
-12222
-11993
-11467
-11332
-10783
-10250
-10114
-10089
-9930
-9434
-9336
-9128
-9109
-8508
-8460
-8198
-8045
-7850
-7342
-7229
-6762
-6302
-6245
-6171
-5957
-5842
-4906
-4904
-4630
-4613
-4567
-4427
-4091
-4084
-3756
-3665
-3367
-3186
-2922
-2372
-2331
-1936
-1683
-1350
-1002
-719
-152
-128
-127
-124
-122
-121
-119
-116
-113
-112
-111
-107
-104
-102
-101
-100
-95
-94
-91
-90
-87
-81
-80
-79
-78
-73
-72
-69
-68
-66
-65
-63
-57
-54
-52
-51
-48
-47
-46
-45
-43
-41
-37
-33
-31
-30
-27
-25
-21
-18
-15
-12
-8
-1
0
1
3
4
5
6
11
14
17
23
25
26
27
28
29
31
32
39
41
46
49
51
52
56
58
61
64
66
67
70
74
79
80
86
88
89
92
93
99
102
104
109
113
117
120
122
123
127
695
912
1792
2857
3150
3184
4060
4626
5671
6412
6827
7999
8017
8646
8798
9703
9837
10049
10442
10912
11400
11430
11436
11551
11937
12480
13258
13469
13689
13963
13982
14019
14152
14259
14346
15416
15613
15954
16241
16814
16844
17564
17702
17751
18537
18763
19890
21216
22238
22548
23243
23383
23386
23407
23940
24076
24796
24870
24898
24967
25139
25176
25699
26167
26536
26614
27008
27087
27142
27356
27458
27800
27827
27924
28595
29053
29229
29884
29900
30460
30556
30701
30815
30995
31613
31761
31772
32099
32308
32674
75627
80472
103073
110477
115718
172418
212268
242652
396135
442591
467087
496849
675960
759343
846297
881562
1003458
1153900
1156733
1164679
1208265
1318372
1363958
1411655
1522329
1559609
1677118
1693658
1703597
1811223
1831642
1838628
1884144
1931545
2085504
2168156
2170263
2239585
2308894
2329235
2364957
2432335
2435551
2596936
2684907
2691011
2705195
2738057
2851897
2925289
2995414
3051534
3216094
3267022
3271559
3338856
3440797
3638325
3651369
3718696
3724814
3811069
3854697
3866969
3893228
3963455
3984546
4098376
4100957
4128113
4200719
4256344
4286332
4306356
4316314
4438803
4458063
4461638
4552228
4563790
4584831
4607992
4884455
4907501
5045419
5066844
5150624
5157161
5190669
5314703
5337397
5434807
5440092
5502665
5523089
5547122
5566200
5582936
5634068
5690330
5776984
5778441
5818505
5826687
5827184
5885735
6010506
6084254
6131498
6138324
6250773
6292801
6306275
6315242
6331640
6484374
6502969
6545970
6666951
6690905
6763576
6766086
6895048
6912227
6929081
6941390
6978168
7045672
7085246
7193307
7197398
7270237
7276767
7295790
7375488
7472098
7687424
7840758
7880957
7904499
7948678
7974126
8015691
8037685
8112955
8131380
8140556
8142384
8220436
8308817
8331317
22581970
45809129
48103779
78212045
79674901
97299830
110308649
131744428
136663461
138485719
139842794
152061792
152685704
153070991
156228213
164884737
174776199
189346581
193148547
208582124
223891881
227308187
237373798
241214067
242476929
245495851
260606593
275202667
285717038
317009689
322759532
325369206
339724742
340122632
345010859
352375176
355826263
359695034
366118516
370008270
382712922
386379440
401153345
404986391
426084981
442843409
473909474
475192613
492302667
494747879
506279889
509813998
537558350
548423414
548467404
566383324
574188786
574792333
591678147
596558084
597423476
602432742
603067874
629552047
630893263
635249783
644959276
650710927
664859367
669433203
684329599
699991513
714451929
723556530
739294558
750895264
757618344
781123405
796973385
801637715
804776709
829003666
829219068
840167037
854882202
860066192
864304878
864808449
867107161
871871263
880591851
883020336
883178082
920223781
936008673
939417822
956776353
958281059
962183717
964059257
967860573
974322643
974999286
980009921
1032949015
1044249483
1064892676
1075628270
1080848022
1085571657
1173635593
1174809080
1176744978
1209783795
1212074975
1252323507
1254757790
1301450562
1302240953
1314501797
1315121266
1339304157
1364304289
1376260506
1383883477
1395158643
1411117754
1440755058
1448365702
1466814914
1468433821
1490105126
1493912601
1495600372
1509536621
1511014977
1545693948
1548924199
1566583103
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1574097219
1597784674
1610710480
1618324005
1646105187
1649417465
1655649169
1660619384
1668826887
1671093718
1672456990
1673477565
1678638502
1682302139
1682515769
1687920300
1690062315
1706031388
1713660819
1772170709
1778416812
1833443690
1861312062
1876004501
1876358085
1882435551
1916050713
1944906037
1950207172
1951593247
1973638546
1976288281
1994977271
2020053060
2025281195
2029716419
2033980539
2052482076
2058251762
2069273004
2073978021
2111013213
2119886932
2134609957
2140349794
2143934987
</code>```

Here is a Python 3 program that can be used to verify and score a submission:

#!/usr/bin/env python3

import shlex
import os
import sys
import subprocess

try:
    from seriously import Seriously
except:
    print("Unable to load Seriously. Run 'pip3 install seriously' to install it.")
    exit()

if len(sys.argv) < 2:
    print("Usage: python3 {} 'command_to_call_your_program'".format(sys.argv[0]))
    exit()

sys.stdin = open(os.devnull, 'r')

with open('nums.txt') as f:
    nums = [int(x) for x in f]

total = 0

for num in nums:
    p = subprocess.Popen(shlex.split(sys.argv[1]), stdin=subprocess.PIPE, stdout=subprocess.PIPE, universal_newlines=True)
    try:
        out = p.communicate(str(num), timeout=30)[0].strip()
    except subprocess.TimeoutExpired:
        print("Program failed to finish within 30 seconds for {}".format(num))
        exit()
    val = Seriously().eval(out)[0]
    if not len(out) <= len(':'+str(num)):
        print("Failed to do at least as well as naive on {} ({})".format(num, out))
        exit()
    elif val != num or not isinstance(val, int):
        print("Incorrect output: {} instead of {}".format(val, num))
        exit()
    else:
        print("{}: {} ({})".format(num, out, len(out)))
        total += len(out)

print("Total:", total)
```

答:
Python 3, 51 52 57 bytes saved

This program brute forces its way through a significant number of 1-5 character programs, using a limited set of 54 56 instructions and a lot of pruning based on the state of the stack before adding each new instruction. It takes about one minute to run on my laptop. 6 characters would take hours.

```#!/usr/bin/env python3
# -*- coding: cp437 -*-

import os
import sys
import json
import math
import time
import random
import timeit

try:
    from seriously import Seriously,chr_cp437,ord_cp437
except:
    print("Unable to load Seriously. Run 'pip3 install seriously' to install it.")
    exit()

MAXLEN=5
MAX=2**31

future_alphabet=[

    chr_cp437(0x4D), #  (M): pop f,[a], execute f for each element in [a], using the element as a temporary stack, push [a] (similar to map(f,[a])); pop [a]: push max([a])
    chr_cp437(0x52), #  (R): pop f,[a]: call f, using [a] as a temporary stack, push [a] (similar to reduce(f,[a])); pop [a]: push [a][::-1] (reverse); pop a: push [1,2,...,a] (range(1,a+1))
    chr_cp437(0x57), #  (W): loop delimiter: peek top of stack, repeat code in loop while a evaluates to true
    chr_cp437(0x60), #  (`): function literal delimiter, pushes function whose body contains all of the commands until the next `. An implied ` is present at EOF if needed.

]

def next_instruction(program,stack):
    plen = len(program)
    slen = len(stack)
    if plen>0 and program[-1]==chr_cp437(0x0B): # introspect the list on top of the stack
        maybes = set(next_instruction(program[:-1],[stack[0][0]]))
        # print(maybes)
        for i in stack[0][1:]:
            maybes = maybes & set(next_instruction(program[:-1],[i]))
            # print(maybes)
        maybes.discard(chr_cp437(0x72)) # no ranges of lists
        maybes.discard(chr_cp437(0xB9)) # no pascal rows of lists
        return list(maybes)

    candidates = []

    if plen==MAXLEN-1 and slen>1:
        if any(isinstance(x,list) for x in stack):
            return [] # lost cause
        if slen==2 and isinstance( stack[0], int ) and isinstance( stack[1], int ):
            candidates = candidates + [
                chr_cp437(0x2A), #  (*): pop a,b: push a*b; pop a,[b] or [b],a: apply a* to each element in the array; pop [a],[b]: push dot product of [a] and [b] (sum([a[i]*b[i] for i in len(a)])) (shorter is padded with 0s)
                chr_cp437(0x2B), #  (+): pop a,b: push a+b; pop [a],[b]: push [a][b] (append [b] to [a]); pop a,[b] or [b],a: apply a+ to each element in the array
                chr_cp437(0x2D), #  (-): pop a,b: push a-b; pop [a],[b]: push [a]-[b] (all elements of [a] not in [b])
                chr_cp437(0x2F), #  (/): pop a,b: push a/b (float division); pop [a]: rotate [a] right by 1, push [a]
                chr_cp437(0x5C), #  (\): pop a,b: push a/b (integer division); pop [a]: rotate [a] left by 1, push [a]
            ]
            if stack[0]>1 and stack[0]<int(math.pow(MAX,1.0/3))+1 and stack[1]>2 and stack[1]<int(math.log(MAX)/math.log(3))+1:
                candidates.append(chr_cp437(0xFC))  #  (ⁿ): pop a,b: push pow(a,b)
            if stack[0]>2 and stack[0]<1000 and stack[1]>2 and stack[1]<stack[0]:
                candidates = candidates + [
                    chr_cp437(0xDB), #  (█): pop a,b: push C(a,b) (aCb)
                    chr_cp437(0xDC) #  (▄): pop a,b: push P(a,b) (aPb)
                ]
        return candidates

    if plen==MAXLEN-1 and slen==1 and isinstance(stack[0],list):
        candidates = candidates + [
            chr_cp437(0xE3), #  (π): pop [a]: push product([a])
            chr_cp437(0xE4)  #  (Σ): pop [a]: push sum([a])
        ]
        return candidates

    candidates = candidates + [
        chr_cp437(0x30), #  (0): push 0
        chr_cp437(0x31), #  (1): push 1
        chr_cp437(0x32), #  (2): push 2
        chr_cp437(0x33), #  (3): push 3
        chr_cp437(0x34), #  (4): push 4
        chr_cp437(0x35), #  (5): push 5
        chr_cp437(0x36), #  (6): push 6
        chr_cp437(0x37), #  (7): push 7
        chr_cp437(0x38), #  (8): push 8
        chr_cp437(0x39)  #  (9): push 9
    ]

    if plen==0 or (plen>0 and program[-1]!=":"):
        candidates.append(chr_cp437(0x3A)) #  (:): numeric literal delimiter: pushes the longest string of characters in '0123456789+-.ij' as a numeric

    # if plen>0 and program[-1]==":":
    #     candidates.append(chr_cp437(0x2D)) # negative literal coming

    if slen>0:
        candidates.append(chr_cp437(0xB3)) #  (│): duplicate stack ([a,b,c] => [a,b,c,a,b,c])

    if slen>0 and isinstance( stack[0], int ):
        candidates = candidates + [
            chr_cp437(0x44), #  (D): pop a: push a-1; pop [a]: push stddev([a])
            chr_cp437(0x54), #  (T): pop a: push tan(a); pop [a],b,c: set [a][b] to c, push [a]
            chr_cp437(0x65), #  (e): pop a: push exp(a)
            chr_cp437(0x75), #  (u): pop a: push a+1
            chr_cp437(0xA9), #  (⌐): pop a: push a+2
            chr_cp437(0xAA), #  (¬): pop a: push a-2
            chr_cp437(0xAB), #  (½): pop a: push a/2 (float division)
            chr_cp437(0xAC), #  (¼): pop a: push a/4 (float division)
            chr_cp437(0xE7), #  (τ): pop a: push 2*a
            chr_cp437(0xF1), #  (±): pop a: push -a (unary negate)
            chr_cp437(0xFD), #  (²): pop a: push a*a
            chr_cp437(0x7E)  #  (~): pop a: push ~a (unary bitwise negate)
        ]
        candidates.append(chr_cp437(0x3B)) #  (;): pop a: push a,a (duplicates top element)
        if stack[0]>=0:
            if stack[0]<13:
                candidates.append(chr_cp437(0x21)) #  (!): pop a: push a! (factorial(a))
                candidates.append(chr_cp437(0xD1)) #  (╤): pop a: push 10**a
            if stack[0]<32:
                candidates.append(chr_cp437(0xD3)) #  (╙): pop a: push 2**a
                candidates.append(chr_cp437(0xB9)) #  (╣): pop a: push the ath row in Pascal's triangle
            if stack[0]<100:
                candidates.append(chr_cp437(0x46)) #  (F): pop a: push Fib(a) (Fib(0)=0, Fib(1)=Fib(2)=1); pop [a]: push a[0]
                candidates.append(chr_cp437(0x50)) #  (P): pop a: push the a-th prime (zero-indexed)
            if stack[0]<500 and plen<MAXLEN-1:
                candidates.append(chr_cp437(0x72)) #  (r): pop a: push [0,1,...,a-1] (range(0,a))

    if slen>0 and isinstance( stack[0], float ):
        candidates = candidates + [
            chr_cp437(0x4B), #  (K): pop a: push ceil(a)
            chr_cp437(0x4C)  #  (L): pop a: push floor(a)
        ]

    if slen>0 and isinstance( stack[0], list ) and len(stack[0])>0:
        candidates = candidates + [
            chr_cp437(0xE3), #  (π): pop [a]: push product([a])
            chr_cp437(0xE4)  #  (Σ): pop [a]: push sum([a])
        ]
        candidates.append(chr_cp437(0x69)) #  (i): pop [a]: push each element from [a], starting from end (flatten)
        if plen<MAXLEN-2:
            candidates.append(chr_cp437(0x0B)) #  (♂): take the next command and map it over the top of the stack (for example, ♂A is equivalent to `A`M)

    if slen>1:
        candidates.append(chr_cp437(0x6B)) #  (k): pop all elements from stack, convert to list (in the order they were on the stack, starting from the top), and push

    if slen>1 and isinstance( stack[0], int ):
        candidates = candidates + [
            chr_cp437(0x61), #  (a): invert the stack ([a,b,c,d] -> [d,c,b,a])
            chr_cp437(0xC5), #  (┼): duplicate each element on stack ([a,b,c] => [a,a,b,b,c,c])
            chr_cp437(0xC7)  #  (╟): pop a: pop a elements and push a list containing those elements in their original order
        ]
        if stack[0]<100:
            candidates.append(chr_cp437(0xC6)) #  (╞): pop a: make a total copies of each element on stack (3 [a,b,c] -> [a,a,a,b,b,b,c,c,c])

    if slen>1 and isinstance( stack[0], int ) and isinstance( stack[1], int ):
        candidates = candidates + [
            chr_cp437(0x2A), #  (*): pop a,b: push a*b; pop a,[b] or [b],a: apply a* to each element in the array; pop [a],[b]: push dot product of [a] and [b] (sum([a[i]*b[i] for i in len(a)])) (shorter is padded with 0s)
            chr_cp437(0x2B), #  (+): pop a,b: push a+b; pop [a],[b]: push [a][b] (append [b] to [a]); pop a,[b] or [b],a: apply a+ to each element in the array
            chr_cp437(0x2D), #  (-): pop a,b: push a-b; pop [a],[b]: push [a]-[b] (all elements of [a] not in [b])
            chr_cp437(0x2F), #  (/): pop a,b: push a/b (float division); pop [a]: rotate [a] right by 1, push [a]
            chr_cp437(0x5C), #  (\): pop a,b: push a/b (integer division); pop [a]: rotate [a] left by 1, push [a]
        ]
        if stack[0]>1 and stack[0]<int(math.pow(MAX,1.0/3))+1 and stack[1]>2 and stack[1]<int(math.log(MAX)/math.log(3))+1:
            candidates.append(chr_cp437(0xFC))  #  (ⁿ): pop a,b: push pow(a,b)
        if stack[0]>2 and stack[0]<1000 and stack[1]>2 and stack[1]<stack[0]:
            candidates = candidates + [
                chr_cp437(0xDB), #  (█): pop a,b: push C(a,b) (aCb)
                chr_cp437(0xDC) #  (▄): pop a,b: push P(a,b) (aPb)
            ]
        if stack[1]-stack[0]>0 and stack[1]-stack[0]<500:
            candidates.append(chr_cp437(0x78)) #  (x): pop a,b: push [a,b) (range(a,b)); pop [a]: push range(*a)

    if slen>1 and (isinstance( stack[0], list ) or isinstance( stack[1], list )):
        candidates = candidates + [
            chr_cp437(0x2A), #  (*): pop a,b: push a*b; pop a,[b] or [b],a: apply a* to each element in the array; pop [a],[b]: push dot product of [a] and [b] (sum([a[i]*b[i] for i in len(a)])) (shorter is padded with 0s)
            chr_cp437(0x2B), #  (+): pop a,b: push a+b; pop [a],[b]: push [a][b] (append [b] to [a]); pop a,[b] or [b],a: apply a+ to each element in the array
            chr_cp437(0x2D), #  (-): pop a,b: push a-b; pop [a],[b]: push [a]-[b] (all elements of [a] not in [b])
            chr_cp437(0x2F), #  (/): pop a,b: push a/b (float division); pop [a]: rotate [a] right by 1, push [a]
            chr_cp437(0x5C), #  (\): pop a,b: push a/b (integer division); pop [a]: rotate [a] left by 1, push [a]
        ]

    return candidates

# p = "8"+chr_cp437(0x72)+chr_cp437(0x0B)
# res = Seriously().eval(p)
# print(p)
# print(res)
# print(next_instruction(p,res))
# sys.exit()

programs = [[['',None]]]
best = {}

def l(n,force_colon=False):
    if force_colon:
        return ':'+str(n)
    if n>=0 and n<10:
        return str(n)
    elif n<0 and n>-10:
        return str(-n)+'±'
    return ':'+str(n)

def cand(n,rep):
    # print(n,rep,len(rep),l(n),len(l(n)),best[n] if n in best else "")
    if n<=MAX and n>=-MAX and (len(rep)==1 or len(rep)<len(l(n))) and (n not in best or len(rep)<len(best[n])):
        best[n] = rep
        # print("woot")
        return True
    return False

for proglen in range(0,MAXLEN+1):
    print(proglen,"/",MAXLEN)
    if proglen<MAXLEN:
        programs.append([])
    first = ""
    for p in programs[proglen]:
        try:
            if first!=p[0][0]:
                first=p[0][0]
                print(" ",p[0][0],"/",9)
        except:
            pass
        if chr_cp437(0x7E) in p[0]:
            print(p[0])
        now = time.clock()
        try:
            res = Seriously().eval(p[0])
            elapsed = time.clock()-now
            if elapsed>0.0005:
                timed = timeit.timeit('Seriously().eval("'+str(p[0].encode('unicode_escape'))+'")','from seriously import Seriously',number=1000)
                if elapsed>0.01:
                    print("SLOW:")
                    print(p[0])
                    print(res)
                    print(elapsed)
            if chr_cp437(0x7E) in p[0]:
                print(res)
            if len(res)==1 and isinstance( res[0], int ):
                # print("cand",res[0],p[0])
                cand(res[0],p[0])
            p[1] = res
            if proglen<MAXLEN:
                # bail out if the stack is too complex to collapse in time
                if proglen==MAXLEN-1:
                    if len(res)>0 and isinstance( res[0], list ) and not isinstance( res[0][0], int ):
                        continue
                    if len(res)>1 and isinstance( res[0], list ):
                        continue
                    if len(res)>2:
                        continue
                for c in next_instruction(p[0],res):
                    programs[proglen+1].append([p[0]+c,None])
        except:
            pass

# print(programs)

def best_or_l(n,force_colon=False):
    if n in best:
        return best[n]
    if n<0 and -n in best:
        return best[-n]+chr_cp437(0xF1)
    return l(n,force_colon)

# for key,value in sorted(best.items()):
#     # if random.random()<0.01:
#     print(key,value)

score = 0
for i in [-2124910654, -2117700574, -2098186988, -2095671996, -2083075613, -2058271687, -2052250777, -2024215903, -2019485642, -2016095616, -2009838326, -2009173317, -2007673992, -2000014444, -1999825668, -1992515610, -1989566707, -1975506037, -1955473208, -1950731112, -1949886423, -1920624450, -1918465596, -1916287469, -1905036556, -1903956118, -1888944417, -1865221863, -1856600057, -1842797147, -1835637734, -1812631357, -1805740096, -1798015647, -1726688233, -1723609647, -1713776890, -1700307138, -1687644479, -1645515069, -1617635994, -1610444000, -1579053372, -1556891649, -1549652116, -1537732956, -1535180388, -1527162056, -1524851611, -1524658412, -1523244369, -1521379172, -1520191198, -1519035741, -1516978241, -1508892332, -1489938422, -1482102944, -1481823232, -1470621147, -1469145091, -1446844485, -1441790509, -1437843276, -1435359182, -1434186947, -1429816311, -1429393781, -1419752032, -1400387846, -1385152926, -1372620863, -1369257355, -1361933674, -1360480816, -1334846204, -1323741718, -1323660173, -1312800992, -1308824840, -1304658551, -1287829999, -1283767920, -1276288210, -1264275838, -1263965596, -1262866901, -1255421887, -1251428680, -1244825786, -1243200329, -1235305532, -1233977691, -1220537074, -1214100716, -1199414474, -1190725823, -1190401800, -1178717689, -1172378149, -1147869245, -1142875190, -1138538768, -1137864183, -1124917489, -1102987222, -1095920186, -1083001017, -1080902655, -1047122002, -1031842676, -1029877334, -1020849489, -1001684838, -998419619, -990915088, -985235989, -982515261, -979074894, -974195629, -973832940, -965324937, -944246431, -938387588, -933873331, -932692878, -928039285, -927947459, -914008773, -907981688, -906376330, -903502449, -898112547, -887444438, -862658502, -843628573, -822463032, -786051095, -776932426, -776033951, -752042328, -746532472, -743149468, -740225710, -734414418, -725435852, -708101516, -699674783, -694869277, -693246525, -690571518, -689249770, -688581912, -686864294, -681445866, -647992869, -641101583, -631409299, -624686189, -613079884, -593711206, -591688546, -591331185, -574790069, -573024823, -565387051, -565137163, -556338668, -556291492, -541411509, -538932064, -500479857, -482419890, -468050561, -424532545, -420534171, -408741873, -406973874, -387664799, -382084509, -367095703, -352332569, -352195997, -346430007, -324596389, -320119776, -306594578, -305952425, -283718911, -267302378, -243302738, -242955859, -232180029, -225394407, -217418127, -212286453, -208344820, -191300139, -186177744, -175765723, -161763935, -157025501, -140389149, -132298608, -126175769, -70566352, -68748576, -53985905, -52674668, -50228620, -39678495, -19825663, -8349922, -8186722, -8125700, -8073135, -8043230, -7994382, -7874433, -7863624, -7784916, -7782477, -7696343, -7607278, -7531250, -7388060, -7368780, -7367625, -7353084, -7334489, -7281141, -7267149, -7140057, -7119066, -7010389, -6992089, -6975258, -6863360, -6784772, -6741079, -6585985, -6550401, -6520011, -6495144, -6459702, -6294273, -6178342, -6169344, -6139663, -6090928, -6022637, -5992707, -5971334, -5925304, -5880940, -5873707, -5807953, -5703992, -5692895, -5535131, -5488812, -5468330, -5404148, -5290247, -5274221, -5264144, -5234715, -5224048, -5179837, -5084014, -5043990, -5028537, -5011679, -4998333, -4922901, -4880159, -4874060, -4787390, -4749096, -4736217, -4502308, -4480611, -4424319, -4363262, -4349743, -4290050, -4240069, -4239657, -4174072, -4093051, -4045363, -4037689, -4033352, -3839265, -3766343, -3716899, -3680075, -3679053, -3581776, -3539227, -3461158, -3282526, -3205947, -3183427, -3169708, -3166430, -3089822, -3061531, -2947574, -2930733, -2919246, -2872882, -2830770, -2739228, -2713826, -2634018, -2613990, -2525529, -2439507, -2432921, -2376201, -2335005, -2307524, -2265548, -2176176, -2123133, -2108773, -2105934, -2075032, -2073940, -2045837, -2045648, -1978182, -1945769, -1935486, -1881608, -1654650, -1602520, -1506746, -1505294, -1475309, -1457605, -1327259, -1312217, -1178723, -1027439, -880781, -833776, -666675, -643098, -593446, -468772, -450369, -443225, -418164, -402004, -319964, -307400, -279414, -190199, -176644, -66862, -32745, -32686, -32352, -32261, -32035, -31928, -31414, -31308, -30925, -30411, -29503, -29182, -28573, -28500, -28093, -27743, -27716, -27351, -27201, -26834, -25946, -25019, -24469, -24341, -24292, -24151, -23732, -22769, -22242, -22002, -20863, -20762, -20644, -20189, -20009, -19142, -19036, -18980, -18616, -18196, -18123, -17942, -17915, -17601, -17494, -16669, -16417, -16317, -15051, -14796, -14742, -14600, -14443, -14159, -14046, -13860, -13804, -13745, -13634, -13498, -13497, -12688, -12471, -12222, -11993, -11467, -11332, -10783, -10250, -10114, -10089, -9930, -9434, -9336, -9128, -9109, -8508, -8460, -8198, -8045, -7850, -7342, -7229, -6762, -6302, -6245, -6171, -5957, -5842, -4906, -4904, -4630, -4613, -4567, -4427, -4091, -4084, -3756, -3665, -3367, -3186, -2922, -2372, -2331, -1936, -1683, -1350, -1002, -719, -152, -128, -127, -124, -122, -121, -119, -116, -113, -112, -111, -107, -104, -102, -101, -100, -95, -94, -91, -90, -87, -81, -80, -79, -78, -73, -72, -69, -68, -66, -65, -63, -57, -54, -52, -51, -48, -47, -46, -45, -43, -41, -37, -33, -31, -30, -27, -25, -21, -18, -15, -12, -8, -1, 0, 1, 3, 4, 5, 6, 11, 14, 17, 23, 25, 26, 27, 28, 29, 31, 32, 39, 41, 46, 49, 51, 52, 56, 58, 61, 64, 66, 67, 70, 74, 79, 80, 86, 88, 89, 92, 93, 99, 102, 104, 109, 113, 117, 120, 122, 123, 127, 695, 912, 1792, 2857, 3150, 3184, 4060, 4626, 5671, 6412, 6827, 7999, 8017, 8646, 8798, 9703, 9837, 10049, 10442, 10912, 11400, 11430, 11436, 11551, 11937, 12480, 13258, 13469, 13689, 13963, 13982, 14019, 14152, 14259, 14346, 15416, 15613, 15954, 16241, 16814, 16844, 17564, 17702, 17751, 18537, 18763, 19890, 21216, 22238, 22548, 23243, 23383, 23386, 23407, 23940, 24076, 24796, 24870, 24898, 24967, 25139, 25176, 25699, 26167, 26536, 26614, 27008, 27087, 27142, 27356, 27458, 27800, 27827, 27924, 28595, 29053, 29229, 29884, 29900, 30460, 30556, 30701, 30815, 30995, 31613, 31761, 31772, 32099, 32308, 32674, 75627, 80472, 103073, 110477, 115718, 172418, 212268, 242652, 396135, 442591, 467087, 496849, 675960, 759343, 846297, 881562, 1003458, 1153900, 1156733, 1164679, 1208265, 1318372, 1363958, 1411655, 1522329, 1559609, 1677118, 1693658, 1703597, 1811223, 1831642, 1838628, 1884144, 1931545, 2085504, 2168156, 2170263, 2239585, 2308894, 2329235, 2364957, 2432335, 2435551, 2596936, 2684907, 2691011, 2705195, 2738057, 2851897, 2925289, 2995414, 3051534, 3216094, 3267022, 3271559, 3338856, 3440797, 3638325, 3651369, 3718696, 3724814, 3811069, 3854697, 3866969, 3893228, 3963455, 3984546, 4098376, 4100957, 4128113, 4200719, 4256344, 4286332, 4306356, 4316314, 4438803, 4458063, 4461638, 4552228, 4563790, 4584831, 4607992, 4884455, 4907501, 5045419, 5066844, 5150624, 5157161, 5190669, 5314703, 5337397, 5434807, 5440092, 5502665, 5523089, 5547122, 5566200, 5582936, 5634068, 5690330, 5776984, 5778441, 5818505, 5826687, 5827184, 5885735, 6010506, 6084254, 6131498, 6138324, 6250773, 6292801, 6306275, 6315242, 6331640, 6484374, 6502969, 6545970, 6666951, 6690905, 6763576, 6766086, 6895048, 6912227, 6929081, 6941390, 6978168, 7045672, 7085246, 7193307, 7197398, 7270237, 7276767, 7295790, 7375488, 7472098, 7687424, 7840758, 7880957, 7904499, 7948678, 7974126, 8015691, 8037685, 8112955, 8131380, 8140556, 8142384, 8220436, 8308817, 8331317, 22581970, 45809129, 48103779, 78212045, 79674901, 97299830, 110308649, 131744428, 136663461, 138485719, 139842794, 152061792, 152685704, 153070991, 156228213, 164884737, 174776199, 189346581, 193148547, 208582124, 223891881, 227308187, 237373798, 241214067, 242476929, 245495851, 260606593, 275202667, 285717038, 317009689, 322759532, 325369206, 339724742, 340122632, 345010859, 352375176, 355826263, 359695034, 366118516, 370008270, 382712922, 386379440, 401153345, 404986391, 426084981, 442843409, 473909474, 475192613, 492302667, 494747879, 506279889, 509813998, 537558350, 548423414, 548467404, 566383324, 574188786, 574792333, 591678147, 596558084, 597423476, 602432742, 603067874, 629552047, 630893263, 635249783, 644959276, 650710927, 664859367, 669433203, 684329599, 699991513, 714451929, 723556530, 739294558, 750895264, 757618344, 781123405, 796973385, 801637715, 804776709, 829003666, 829219068, 840167037, 854882202, 860066192, 864304878, 864808449, 867107161, 871871263, 880591851, 883020336, 883178082, 920223781, 936008673, 939417822, 956776353, 958281059, 962183717, 964059257, 967860573, 974322643, 974999286, 980009921, 1032949015, 1044249483, 1064892676, 1075628270, 1080848022, 1085571657, 1173635593, 1174809080, 1176744978, 1209783795, 1212074975, 1252323507, 1254757790, 1301450562, 1302240953, 1314501797, 1315121266, 1339304157, 1364304289, 1376260506, 1383883477, 1395158643, 1411117754, 1440755058, 1448365702, 1466814914, 1468433821, 1490105126, 1493912601, 1495600372, 1509536621, 1511014977, 1545693948, 1548924199, 1566583103, 1569747154, 1574097219, 1597784674, 1610710480, 1618324005, 1646105187, 1649417465, 1655649169, 1660619384, 1668826887, 1671093718, 1672456990, 1673477565, 1678638502, 1682302139, 1682515769, 1687920300, 1690062315, 1706031388, 1713660819, 1772170709, 1778416812, 1833443690, 1861312062, 1876004501, 1876358085, 1882435551, 1916050713, 1944906037, 1950207172, 1951593247, 1973638546, 1976288281, 1994977271, 2020053060, 2025281195, 2029716419, 2033980539, 2052482076, 2058251762, 2069273004, 2073978021, 2111013213, 2119886932, 2134609957, 2140349794, 2143934987]:
    print(i,best_or_l(i,True),len(l(i)),len(best_or_l(i,True)))
    score = score + (len(l(i))-len(best_or_l(i,True)))
print("Score:",score)

with open("brute_serious.json","w") as outfile:
    outfile.write(json.dumps(best,sort_keys=True,indent=2))
```

<p>Here's a random sampling of the sorts of Actually programs that end up found by this algorithm:</p>

```-387420483 99ⁿ6-
-16777208 4!╙8-
-999999 6╤1-
-362864 9!4²-
-46369 4!Fu±
-5045 57!±-
8101 9eL¬
19900 2╤τrΣ
46367 4!FD
68072 5╙│F\
99990 1╤5╤-
156246 75ⁿ¬τ
518393 76!²-
1814399 59!*D
6534927 35╙F*
14930357 56²F+
19999995 57╤τ-
25396560 7!│D*
65691025 9eKu²
100001000 3╤8╤+
200000002 8╤uτ
800000008 88╤u*
1626347584 88!+²
2000000018 99╤+τ
```

Title: Actually Integer Metagolf

Body:
Python 3, 51 52 57 bytes saved

This program brute forces its way through a significant number of 1-5 character programs, using a limited set of 54 56 instructions and a lot of pruning based on the state of the stack before adding each new instruction. It takes about one minute to run on my laptop. 6 characters would take hours.

#!/usr/bin/env python3
# -*- coding: cp437 -*-

import os
import sys
import json
import math
import time
import random
import timeit

try:
    from seriously import Seriously,chr_cp437,ord_cp437
except:
    print("Unable to load Seriously. Run 'pip3 install seriously' to install it.")
    exit()

MAXLEN=5
MAX=2**31

future_alphabet=[

    chr_cp437(0x4D), #  (M): pop f,[a], execute f for each element in [a], using the element as a temporary stack, push [a] (similar to map(f,[a])); pop [a]: push max([a])
    chr_cp437(0x52), #  (R): pop f,[a]: call f, using [a] as a temporary stack, push [a] (similar to reduce(f,[a])); pop [a]: push [a][::-1] (reverse); pop a: push [1,2,...,a] (range(1,a+1))
    chr_cp437(0x57), #  (W): loop delimiter: peek top of stack, repeat code in loop while a evaluates to true
    chr_cp437(0x60), #  (`): function literal delimiter, pushes function whose body contains all of the commands until the next `. An implied ` is present at EOF if needed.

]

def next_instruction(program,stack):
    plen = len(program)
    slen = len(stack)
    if plen>0 and program[-1]==chr_cp437(0x0B): # introspect the list on top of the stack
        maybes = set(next_instruction(program[:-1],[stack[0][0]]))
        # print(maybes)
        for i in stack[0][1:]:
            maybes = maybes & set(next_instruction(program[:-1],[i]))
            # print(maybes)
        maybes.discard(chr_cp437(0x72)) # no ranges of lists
        maybes.discard(chr_cp437(0xB9)) # no pascal rows of lists
        return list(maybes)

    candidates = []

    if plen==MAXLEN-1 and slen>1:
        if any(isinstance(x,list) for x in stack):
            return [] # lost cause
        if slen==2 and isinstance( stack[0], int ) and isinstance( stack[1], int ):
            candidates = candidates + [
                chr_cp437(0x2A), #  (*): pop a,b: push a*b; pop a,[b] or [b],a: apply a* to each element in the array; pop [a],[b]: push dot product of [a] and [b] (sum([a[i]*b[i] for i in len(a)])) (shorter is padded with 0s)
                chr_cp437(0x2B), #  (+): pop a,b: push a+b; pop [a],[b]: push [a][b] (append [b] to [a]); pop a,[b] or [b],a: apply a+ to each element in the array
                chr_cp437(0x2D), #  (-): pop a,b: push a-b; pop [a],[b]: push [a]-[b] (all elements of [a] not in [b])
                chr_cp437(0x2F), #  (/): pop a,b: push a/b (float division); pop [a]: rotate [a] right by 1, push [a]
                chr_cp437(0x5C), #  (\): pop a,b: push a/b (integer division); pop [a]: rotate [a] left by 1, push [a]
            ]
            if stack[0]>1 and stack[0]<int(math.pow(MAX,1.0/3))+1 and stack[1]>2 and stack[1]<int(math.log(MAX)/math.log(3))+1:
                candidates.append(chr_cp437(0xFC))  #  (ⁿ): pop a,b: push pow(a,b)
            if stack[0]>2 and stack[0]<1000 and stack[1]>2 and stack[1]<stack[0]:
                candidates = candidates + [
                    chr_cp437(0xDB), #  (█): pop a,b: push C(a,b) (aCb)
                    chr_cp437(0xDC) #  (▄): pop a,b: push P(a,b) (aPb)
                ]
        return candidates

    if plen==MAXLEN-1 and slen==1 and isinstance(stack[0],list):
        candidates = candidates + [
            chr_cp437(0xE3), #  (π): pop [a]: push product([a])
            chr_cp437(0xE4)  #  (Σ): pop [a]: push sum([a])
        ]
        return candidates

    candidates = candidates + [
        chr_cp437(0x30), #  (0): push 0
        chr_cp437(0x31), #  (1): push 1
        chr_cp437(0x32), #  (2): push 2
        chr_cp437(0x33), #  (3): push 3
        chr_cp437(0x34), #  (4): push 4
        chr_cp437(0x35), #  (5): push 5
        chr_cp437(0x36), #  (6): push 6
        chr_cp437(0x37), #  (7): push 7
        chr_cp437(0x38), #  (8): push 8
        chr_cp437(0x39)  #  (9): push 9
    ]

    if plen==0 or (plen>0 and program[-1]!=":"):
        candidates.append(chr_cp437(0x3A)) #  (:): numeric literal delimiter: pushes the longest string of characters in '0123456789+-.ij' as a numeric

    # if plen>0 and program[-1]==":":
    #     candidates.append(chr_cp437(0x2D)) # negative literal coming

    if slen>0:
        candidates.append(chr_cp437(0xB3)) #  (│): duplicate stack ([a,b,c] => [a,b,c,a,b,c])

    if slen>0 and isinstance( stack[0], int ):
        candidates = candidates + [
            chr_cp437(0x44), #  (D): pop a: push a-1; pop [a]: push stddev([a])
            chr_cp437(0x54), #  (T): pop a: push tan(a); pop [a],b,c: set [a][b] to c, push [a]
            chr_cp437(0x65), #  (e): pop a: push exp(a)
            chr_cp437(0x75), #  (u): pop a: push a+1
            chr_cp437(0xA9), #  (⌐): pop a: push a+2
            chr_cp437(0xAA), #  (¬): pop a: push a-2
            chr_cp437(0xAB), #  (½): pop a: push a/2 (float division)
            chr_cp437(0xAC), #  (¼): pop a: push a/4 (float division)
            chr_cp437(0xE7), #  (τ): pop a: push 2*a
            chr_cp437(0xF1), #  (±): pop a: push -a (unary negate)
            chr_cp437(0xFD), #  (²): pop a: push a*a
            chr_cp437(0x7E)  #  (~): pop a: push ~a (unary bitwise negate)
        ]
        candidates.append(chr_cp437(0x3B)) #  (;): pop a: push a,a (duplicates top element)
        if stack[0]>=0:
            if stack[0]<13:
                candidates.append(chr_cp437(0x21)) #  (!): pop a: push a! (factorial(a))
                candidates.append(chr_cp437(0xD1)) #  (╤): pop a: push 10**a
            if stack[0]<32:
                candidates.append(chr_cp437(0xD3)) #  (╙): pop a: push 2**a
                candidates.append(chr_cp437(0xB9)) #  (╣): pop a: push the ath row in Pascal's triangle
            if stack[0]<100:
                candidates.append(chr_cp437(0x46)) #  (F): pop a: push Fib(a) (Fib(0)=0, Fib(1)=Fib(2)=1); pop [a]: push a[0]
                candidates.append(chr_cp437(0x50)) #  (P): pop a: push the a-th prime (zero-indexed)
            if stack[0]<500 and plen<MAXLEN-1:
                candidates.append(chr_cp437(0x72)) #  (r): pop a: push [0,1,...,a-1] (range(0,a))

    if slen>0 and isinstance( stack[0], float ):
        candidates = candidates + [
            chr_cp437(0x4B), #  (K): pop a: push ceil(a)
            chr_cp437(0x4C)  #  (L): pop a: push floor(a)
        ]

    if slen>0 and isinstance( stack[0], list ) and len(stack[0])>0:
        candidates = candidates + [
            chr_cp437(0xE3), #  (π): pop [a]: push product([a])
            chr_cp437(0xE4)  #  (Σ): pop [a]: push sum([a])
        ]
        candidates.append(chr_cp437(0x69)) #  (i): pop [a]: push each element from [a], starting from end (flatten)
        if plen<MAXLEN-2:
            candidates.append(chr_cp437(0x0B)) #  (♂): take the next command and map it over the top of the stack (for example, ♂A is equivalent to `A`M)

    if slen>1:
        candidates.append(chr_cp437(0x6B)) #  (k): pop all elements from stack, convert to list (in the order they were on the stack, starting from the top), and push

    if slen>1 and isinstance( stack[0], int ):
        candidates = candidates + [
            chr_cp437(0x61), #  (a): invert the stack ([a,b,c,d] -> [d,c,b,a])
            chr_cp437(0xC5), #  (┼): duplicate each element on stack ([a,b,c] => [a,a,b,b,c,c])
            chr_cp437(0xC7)  #  (╟): pop a: pop a elements and push a list containing those elements in their original order
        ]
        if stack[0]<100:
            candidates.append(chr_cp437(0xC6)) #  (╞): pop a: make a total copies of each element on stack (3 [a,b,c] -> [a,a,a,b,b,b,c,c,c])

    if slen>1 and isinstance( stack[0], int ) and isinstance( stack[1], int ):
        candidates = candidates + [
            chr_cp437(0x2A), #  (*): pop a,b: push a*b; pop a,[b] or [b],a: apply a* to each element in the array; pop [a],[b]: push dot product of [a] and [b] (sum([a[i]*b[i] for i in len(a)])) (shorter is padded with 0s)
            chr_cp437(0x2B), #  (+): pop a,b: push a+b; pop [a],[b]: push [a][b] (append [b] to [a]); pop a,[b] or [b],a: apply a+ to each element in the array
            chr_cp437(0x2D), #  (-): pop a,b: push a-b; pop [a],[b]: push [a]-[b] (all elements of [a] not in [b])
            chr_cp437(0x2F), #  (/): pop a,b: push a/b (float division); pop [a]: rotate [a] right by 1, push [a]
            chr_cp437(0x5C), #  (\): pop a,b: push a/b (integer division); pop [a]: rotate [a] left by 1, push [a]
        ]
        if stack[0]>1 and stack[0]<int(math.pow(MAX,1.0/3))+1 and stack[1]>2 and stack[1]<int(math.log(MAX)/math.log(3))+1:
            candidates.append(chr_cp437(0xFC))  #  (ⁿ): pop a,b: push pow(a,b)
        if stack[0]>2 and stack[0]<1000 and stack[1]>2 and stack[1]<stack[0]:
            candidates = candidates + [
                chr_cp437(0xDB), #  (█): pop a,b: push C(a,b) (aCb)
                chr_cp437(0xDC) #  (▄): pop a,b: push P(a,b) (aPb)
            ]
        if stack[1]-stack[0]>0 and stack[1]-stack[0]<500:
            candidates.append(chr_cp437(0x78)) #  (x): pop a,b: push [a,b) (range(a,b)); pop [a]: push range(*a)

    if slen>1 and (isinstance( stack[0], list ) or isinstance( stack[1], list )):
        candidates = candidates + [
            chr_cp437(0x2A), #  (*): pop a,b: push a*b; pop a,[b] or [b],a: apply a* to each element in the array; pop [a],[b]: push dot product of [a] and [b] (sum([a[i]*b[i] for i in len(a)])) (shorter is padded with 0s)
            chr_cp437(0x2B), #  (+): pop a,b: push a+b; pop [a],[b]: push [a][b] (append [b] to [a]); pop a,[b] or [b],a: apply a+ to each element in the array
            chr_cp437(0x2D), #  (-): pop a,b: push a-b; pop [a],[b]: push [a]-[b] (all elements of [a] not in [b])
            chr_cp437(0x2F), #  (/): pop a,b: push a/b (float division); pop [a]: rotate [a] right by 1, push [a]
            chr_cp437(0x5C), #  (\): pop a,b: push a/b (integer division); pop [a]: rotate [a] left by 1, push [a]
        ]

    return candidates

# p = "8"+chr_cp437(0x72)+chr_cp437(0x0B)
# res = Seriously().eval(p)
# print(p)
# print(res)
# print(next_instruction(p,res))
# sys.exit()

programs = [[['',None]]]
best = {}

def l(n,force_colon=False):
    if force_colon:
        return ':'+str(n)
    if n>=0 and n<10:
        return str(n)
    elif n<0 and n>-10:
        return str(-n)+'±'
    return ':'+str(n)

def cand(n,rep):
    # print(n,rep,len(rep),l(n),len(l(n)),best[n] if n in best else "")
    if n<=MAX and n>=-MAX and (len(rep)==1 or len(rep)<len(l(n))) and (n not in best or len(rep)<len(best[n])):
        best[n] = rep
        # print("woot")
        return True
    return False

for proglen in range(0,MAXLEN+1):
    print(proglen,"/",MAXLEN)
    if proglen<MAXLEN:
        programs.append([])
    first = ""
    for p in programs[proglen]:
        try:
            if first!=p[0][0]:
                first=p[0][0]
                print(" ",p[0][0],"/",9)
        except:
            pass
        if chr_cp437(0x7E) in p[0]:
            print(p[0])
        now = time.clock()
        try:
            res = Seriously().eval(p[0])
            elapsed = time.clock()-now
            if elapsed>0.0005:
                timed = timeit.timeit('Seriously().eval("'+str(p[0].encode('unicode_escape'))+'")','from seriously import Seriously',number=1000)
                if elapsed>0.01:
                    print("SLOW:")
                    print(p[0])
                    print(res)
                    print(elapsed)
            if chr_cp437(0x7E) in p[0]:
                print(res)
            if len(res)==1 and isinstance( res[0], int ):
                # print("cand",res[0],p[0])
                cand(res[0],p[0])
            p[1] = res
            if proglen<MAXLEN:
                # bail out if the stack is too complex to collapse in time
                if proglen==MAXLEN-1:
                    if len(res)>0 and isinstance( res[0], list ) and not isinstance( res[0][0], int ):
                        continue
                    if len(res)>1 and isinstance( res[0], list ):
                        continue
                    if len(res)>2:
                        continue
                for c in next_instruction(p[0],res):
                    programs[proglen+1].append([p[0]+c,None])
        except:
            pass

# print(programs)

def best_or_l(n,force_colon=False):
    if n in best:
        return best[n]
    if n<0 and -n in best:
        return best[-n]+chr_cp437(0xF1)
    return l(n,force_colon)

# for key,value in sorted(best.items()):
#     # if random.random()<0.01:
#     print(key,value)

score = 0
for i in [-2124910654, -2117700574, -2098186988, -2095671996, -2083075613, -2058271687, -2052250777, -2024215903, -2019485642, -2016095616, -2009838326, -2009173317, -2007673992, -2000014444, -1999825668, -1992515610, -1989566707, -1975506037, -1955473208, -1950731112, -1949886423, -1920624450, -1918465596, -1916287469, -1905036556, -1903956118, -1888944417, -1865221863, -1856600057, -1842797147, -1835637734, -1812631357, -1805740096, -1798015647, -1726688233, -1723609647, -1713776890, -1700307138, -1687644479, -1645515069, -1617635994, -1610444000, -1579053372, -1556891649, -1549652116, -1537732956, -1535180388, -1527162056, -1524851611, -1524658412, -1523244369, -1521379172, -1520191198, -1519035741, -1516978241, -1508892332, -1489938422, -1482102944, -1481823232, -1470621147, -1469145091, -1446844485, -1441790509, -1437843276, -1435359182, -1434186947, -1429816311, -1429393781, -1419752032, -1400387846, -1385152926, -1372620863, -1369257355, -1361933674, -1360480816, -1334846204, -1323741718, -1323660173, -1312800992, -1308824840, -1304658551, -1287829999, -1283767920, -1276288210, -1264275838, -1263965596, -1262866901, -1255421887, -1251428680, -1244825786, -1243200329, -1235305532, -1233977691, -1220537074, -1214100716, -1199414474, -1190725823, -1190401800, -1178717689, -1172378149, -1147869245, -1142875190, -1138538768, -1137864183, -1124917489, -1102987222, -1095920186, -1083001017, -1080902655, -1047122002, -1031842676, -1029877334, -1020849489, -1001684838, -998419619, -990915088, -985235989, -982515261, -979074894, -974195629, -973832940, -965324937, -944246431, -938387588, -933873331, -932692878, -928039285, -927947459, -914008773, -907981688, -906376330, -903502449, -898112547, -887444438, -862658502, -843628573, -822463032, -786051095, -776932426, -776033951, -752042328, -746532472, -743149468, -740225710, -734414418, -725435852, -708101516, -699674783, -694869277, -693246525, -690571518, -689249770, -688581912, -686864294, -681445866, -647992869, -641101583, -631409299, -624686189, -613079884, -593711206, -591688546, -591331185, -574790069, -573024823, -565387051, -565137163, -556338668, -556291492, -541411509, -538932064, -500479857, -482419890, -468050561, -424532545, -420534171, -408741873, -406973874, -387664799, -382084509, -367095703, -352332569, -352195997, -346430007, -324596389, -320119776, -306594578, -305952425, -283718911, -267302378, -243302738, -242955859, -232180029, -225394407, -217418127, -212286453, -208344820, -191300139, -186177744, -175765723, -161763935, -157025501, -140389149, -132298608, -126175769, -70566352, -68748576, -53985905, -52674668, -50228620, -39678495, -19825663, -8349922, -8186722, -8125700, -8073135, -8043230, -7994382, -7874433, -7863624, -7784916, -7782477, -7696343, -7607278, -7531250, -7388060, -7368780, -7367625, -7353084, -7334489, -7281141, -7267149, -7140057, -7119066, -7010389, -6992089, -6975258, -6863360, -6784772, -6741079, -6585985, -6550401, -6520011, -6495144, -6459702, -6294273, -6178342, -6169344, -6139663, -6090928, -6022637, -5992707, -5971334, -5925304, -5880940, -5873707, -5807953, -5703992, -5692895, -5535131, -5488812, -5468330, -5404148, -5290247, -5274221, -5264144, -5234715, -5224048, -5179837, -5084014, -5043990, -5028537, -5011679, -4998333, -4922901, -4880159, -4874060, -4787390, -4749096, -4736217, -4502308, -4480611, -4424319, -4363262, -4349743, -4290050, -4240069, -4239657, -4174072, -4093051, -4045363, -4037689, -4033352, -3839265, -3766343, -3716899, -3680075, -3679053, -3581776, -3539227, -3461158, -3282526, -3205947, -3183427, -3169708, -3166430, -3089822, -3061531, -2947574, -2930733, -2919246, -2872882, -2830770, -2739228, -2713826, -2634018, -2613990, -2525529, -2439507, -2432921, -2376201, -2335005, -2307524, -2265548, -2176176, -2123133, -2108773, -2105934, -2075032, -2073940, -2045837, -2045648, -1978182, -1945769, -1935486, -1881608, -1654650, -1602520, -1506746, -1505294, -1475309, -1457605, -1327259, -1312217, -1178723, -1027439, -880781, -833776, -666675, -643098, -593446, -468772, -450369, -443225, -418164, -402004, -319964, -307400, -279414, -190199, -176644, -66862, -32745, -32686, -32352, -32261, -32035, -31928, -31414, -31308, -30925, -30411, -29503, -29182, -28573, -28500, -28093, -27743, -27716, -27351, -27201, -26834, -25946, -25019, -24469, -24341, -24292, -24151, -23732, -22769, -22242, -22002, -20863, -20762, -20644, -20189, -20009, -19142, -19036, -18980, -18616, -18196, -18123, -17942, -17915, -17601, -17494, -16669, -16417, -16317, -15051, -14796, -14742, -14600, -14443, -14159, -14046, -13860, -13804, -13745, -13634, -13498, -13497, -12688, -12471, -12222, -11993, -11467, -11332, -10783, -10250, -10114, -10089, -9930, -9434, -9336, -9128, -9109, -8508, -8460, -8198, -8045, -7850, -7342, -7229, -6762, -6302, -6245, -6171, -5957, -5842, -4906, -4904, -4630, -4613, -4567, -4427, -4091, -4084, -3756, -3665, -3367, -3186, -2922, -2372, -2331, -1936, -1683, -1350, -1002, -719, -152, -128, -127, -124, -122, -121, -119, -116, -113, -112, -111, -107, -104, -102, -101, -100, -95, -94, -91, -90, -87, -81, -80, -79, -78, -73, -72, -69, -68, -66, -65, -63, -57, -54, -52, -51, -48, -47, -46, -45, -43, -41, -37, -33, -31, -30, -27, -25, -21, -18, -15, -12, -8, -1, 0, 1, 3, 4, 5, 6, 11, 14, 17, 23, 25, 26, 27, 28, 29, 31, 32, 39, 41, 46, 49, 51, 52, 56, 58, 61, 64, 66, 67, 70, 74, 79, 80, 86, 88, 89, 92, 93, 99, 102, 104, 109, 113, 117, 120, 122, 123, 127, 695, 912, 1792, 2857, 3150, 3184, 4060, 4626, 5671, 6412, 6827, 7999, 8017, 8646, 8798, 9703, 9837, 10049, 10442, 10912, 11400, 11430, 11436, 11551, 11937, 12480, 13258, 13469, 13689, 13963, 13982, 14019, 14152, 14259, 14346, 15416, 15613, 15954, 16241, 16814, 16844, 17564, 17702, 17751, 18537, 18763, 19890, 21216, 22238, 22548, 23243, 23383, 23386, 23407, 23940, 24076, 24796, 24870, 24898, 24967, 25139, 25176, 25699, 26167, 26536, 26614, 27008, 27087, 27142, 27356, 27458, 27800, 27827, 27924, 28595, 29053, 29229, 29884, 29900, 30460, 30556, 30701, 30815, 30995, 31613, 31761, 31772, 32099, 32308, 32674, 75627, 80472, 103073, 110477, 115718, 172418, 212268, 242652, 396135, 442591, 467087, 496849, 675960, 759343, 846297, 881562, 1003458, 1153900, 1156733, 1164679, 1208265, 1318372, 1363958, 1411655, 1522329, 1559609, 1677118, 1693658, 1703597, 1811223, 1831642, 1838628, 1884144, 1931545, 2085504, 2168156, 2170263, 2239585, 2308894, 2329235, 2364957, 2432335, 2435551, 2596936, 2684907, 2691011, 2705195, 2738057, 2851897, 2925289, 2995414, 3051534, 3216094, 3267022, 3271559, 3338856, 3440797, 3638325, 3651369, 3718696, 3724814, 3811069, 3854697, 3866969, 3893228, 3963455, 3984546, 4098376, 4100957, 4128113, 4200719, 4256344, 4286332, 4306356, 4316314, 4438803, 4458063, 4461638, 4552228, 4563790, 4584831, 4607992, 4884455, 4907501, 5045419, 5066844, 5150624, 5157161, 5190669, 5314703, 5337397, 5434807, 5440092, 5502665, 5523089, 5547122, 5566200, 5582936, 5634068, 5690330, 5776984, 5778441, 5818505, 5826687, 5827184, 5885735, 6010506, 6084254, 6131498, 6138324, 6250773, 6292801, 6306275, 6315242, 6331640, 6484374, 6502969, 6545970, 6666951, 6690905, 6763576, 6766086, 6895048, 6912227, 6929081, 6941390, 6978168, 7045672, 7085246, 7193307, 7197398, 7270237, 7276767, 7295790, 7375488, 7472098, 7687424, 7840758, 7880957, 7904499, 7948678, 7974126, 8015691, 8037685, 8112955, 8131380, 8140556, 8142384, 8220436, 8308817, 8331317, 22581970, 45809129, 48103779, 78212045, 79674901, 97299830, 110308649, 131744428, 136663461, 138485719, 139842794, 152061792, 152685704, 153070991, 156228213, 164884737, 174776199, 189346581, 193148547, 208582124, 223891881, 227308187, 237373798, 241214067, 242476929, 245495851, 260606593, 275202667, 285717038, 317009689, 322759532, 325369206, 339724742, 340122632, 345010859, 352375176, 355826263, 359695034, 366118516, 370008270, 382712922, 386379440, 401153345, 404986391, 426084981, 442843409, 473909474, 475192613, 492302667, 494747879, 506279889, 509813998, 537558350, 548423414, 548467404, 566383324, 574188786, 574792333, 591678147, 596558084, 597423476, 602432742, 603067874, 629552047, 630893263, 635249783, 644959276, 650710927, 664859367, 669433203, 684329599, 699991513, 714451929, 723556530, 739294558, 750895264, 757618344, 781123405, 796973385, 801637715, 804776709, 829003666, 829219068, 840167037, 854882202, 860066192, 864304878, 864808449, 867107161, 871871263, 880591851, 883020336, 883178082, 920223781, 936008673, 939417822, 956776353, 958281059, 962183717, 964059257, 967860573, 974322643, 974999286, 980009921, 1032949015, 1044249483, 1064892676, 1075628270, 1080848022, 1085571657, 1173635593, 1174809080, 1176744978, 1209783795, 1212074975, 1252323507, 1254757790, 1301450562, 1302240953, 1314501797, 1315121266, 1339304157, 1364304289, 1376260506, 1383883477, 1395158643, 1411117754, 1440755058, 1448365702, 1466814914, 1468433821, 1490105126, 1493912601, 1495600372, 1509536621, 1511014977, 1545693948, 1548924199, 1566583103, 1569747154, 1574097219, 1597784674, 1610710480, 1618324005, 1646105187, 1649417465, 1655649169, 1660619384, 1668826887, 1671093718, 1672456990, 1673477565, 1678638502, 1682302139, 1682515769, 1687920300, 1690062315, 1706031388, 1713660819, 1772170709, 1778416812, 1833443690, 1861312062, 1876004501, 1876358085, 1882435551, 1916050713, 1944906037, 1950207172, 1951593247, 1973638546, 1976288281, 1994977271, 2020053060, 2025281195, 2029716419, 2033980539, 2052482076, 2058251762, 2069273004, 2073978021, 2111013213, 2119886932, 2134609957, 2140349794, 2143934987]:
    print(i,best_or_l(i,True),len(l(i)),len(best_or_l(i,True)))
    score = score + (len(l(i))-len(best_or_l(i,True)))
print("Score:",score)

with open("brute_serious.json","w") as outfile:
    outfile.write(json.dumps(best,sort_keys=True,indent=2))

Here's a random sampling of the sorts of Actually programs that end up found by this algorithm:

-387420483 99ⁿ6-
-16777208 4!╙8-
-999999 6╤1-
-362864 9!4²-
-46369 4!Fu±
-5045 57!±-
8101 9eL¬
19900 2╤τrΣ
46367 4!FD
68072 5╙│F\
99990 1╤5╤-
156246 75ⁿ¬τ
518393 76!²-
1814399 59!*D
6534927 35╙F*
14930357 56²F+
19999995 57╤τ-
25396560 7!│D*
65691025 9eKu²
100001000 3╤8╤+
200000002 8╤uτ
800000008 88╤u*
1626347584 88!+²
2000000018 99╤+τ


问:
Golf Text into DNA

Text to DNA golf

Challenge

Convert input into a DNA output.

Algorithm

Convert text into ASCII code points (e.g. codegolf -> [99, 111, 100, 101, 103, 111, 108, 102])

String the ASCII codes together (e.g. 99111100101103111108102)

Convert to binary (e.g. 10100111111001101001011010001000011001101011011110000110010111111011000000110)

Pad 0s onto the end to make an even number of characters (e.g. 101001111110011010010110100010000110011010110111100001100101111110110000001100)

Replace 00 with A, 01 with C, 10 with G, and 11 with T (e.g. GGCTTGCGGCCGGAGACGCGGTCTGACGCCTTGTAAATA)

Output

Test Cases

```<code>codegolf > GGCTTGCGGCCGGAGACGCGGTCTGACGCCTTGTAAATA
ppcg > GGCTAATTGTCGCACTT
} > TTGG (padding)
</code>```

Specifications

This is code-golf

Your program must accept spaces in input.

Your program must work for codegolf.

答:
Perl 6, 57 + 1 (-p flag) = 58 bytes

```<code>$_=(+[~] .ords).base(2);s:g/..?/{<A G C T>[:2($/.flip)]}/
</code>```

Step by step explanation:

-p flag causes Perl 6 interpreter to run code line by line, put current line $_, and at end put it back from $_.

.ords - If there is nothing before a period, a method is called on $_. ords method returns list of codepoints in a string.

[~] - [] is a reduction operator, which stores its reduction operator between brackets. In this case, it's ~, which is a string concatenation operator. For example, [~] 1, 2, 3 is equivalent to 1 ~ 2 ~ 3.

+ converts its argument to a number, needed because base method is only defined for integers.

.base(2) - converts an integer to a string in base 2

$_= - assigns the result to $_.

s:g/..?/{...}/ - this is a regular expression replacing any (:g, global mode) instance of regex ..? (one or two characters). The second argument is a replacement pattern, which in this case in code (in Perl 6, curly brackets in strings and replacement patterns are executed as code).

$/ - a regex match variable

.flip - inverts a string. It implicitly converts $/ (a regex match object) to a string. This is because a single character 1 should be expanded to 10, as opposed to 01. Because of that flip, order of elements in array has G and C reversed.

:2(...) - parses a base-2 string into an integer.

 - array of four elements.

...[...] - array access operator.

What does that mean? The program gets list of all codepoints in a string, concatenates them together, converts them to base 2. Then, it replaces all instances of two or one character into one of letters A, G, C, T depending on flipped representation of a number in binary.

ContentLicense: CC BY-SA 3.0

Title: Golf Text into DNA

Body:
Perl 6, 57 + 1 (-p flag) = 58 bytes

$_=(+[~] .ords).base(2);s:g/..?/{<A G C T>[:2($/.flip)]}/

Step by step explanation:

-p flag causes Perl 6 interpreter to run code line by line, put current line $_, and at end put it back from $_.

.ords - If there is nothing before a period, a method is called on $_. ords method returns list of codepoints in a string.

[~] - [] is a reduction operator, which stores its reduction operator between brackets. In this case, it's ~, which is a string concatenation operator. For example, [~] 1, 2, 3 is equivalent to 1 ~ 2 ~ 3.

+ converts its argument to a number, needed because base method is only defined for integers.

.base(2) - converts an integer to a string in base 2

$_= - assigns the result to $_.

s:g/..?/{...}/ - this is a regular expression replacing any (:g, global mode) instance of regex ..? (one or two characters). The second argument is a replacement pattern, which in this case in code (in Perl 6, curly brackets in strings and replacement patterns are executed as code).

$/ - a regex match variable

.flip - inverts a string. It implicitly converts $/ (a regex match object) to a string. This is because a single character 1 should be expanded to 10, as opposed to 01. Because of that flip, order of elements in array has G and C reversed.

:2(...) - parses a base-2 string into an integer.

<A G C T> - array of four elements.

...[...] - array access operator.

What does that mean? The program gets list of all codepoints in a string, concatenates them together, converts them to base 2. Then, it replaces all instances of two or one character into one of letters A, G, C, T depending on flipped representation of a number in binary.


问:
How should you arrange your chairs?
You teach a class of students with interesting preferences for how their chairs are arranged. There are 3 very specific requirements they have for how the chairs are arranged:

They most be arranged in a rectangle, even if it means some chairs go empty.

There must be as few empty chairs as possible.

They must be as "squarey" as possible. Squarey-ness is determined by the distance between the width and the height of the rectangle, lower is better. For example, a rectangle that is 4x7 has a squarey-ness of 3.

To be more specific, the "score" of an arrangement is the distance between the width and height plus the number of chairs that would go empty.

Let's take an example. Let's say you have 13 students. You could arrange the chairs any of these ways:

```<code>1x13
2x7
3x5
4x4
</code>```

1x13 is not very squarey. In fact, 1 and 13 are 12 apart, so we give this arrangement 12 points. It also has 0 empty chairs, so we add 0 points, giving this arrangement a score of 12. Not that great.

2x7 is certainly better. 2 and 7 are only 5 apart, so we give this arrangement 5 points. However, if you actually arranged 2 rows of seven chairs, that would take 14 chairs, meaning one chair would be empty. So we add one point, giving this arrangement a score of 6.

We could also do 3x5. 3 and 5 are 2 apart, so +2 points. It takes 15 chairs, meaning we'd have two extra chairs, so another +2 points, for a score of 4.

Last option, 4x4. 4 and 4 are 0 apart, so we give this +0 points. 4x4 takes 16 chairs, so 3 chairs go empty, for a total score of 3. This is the optimal solution.

In case of a tie, the optimal solution is the one with less empty chairs.

The challenge

You must write a program or function that takes an integer and outputs the optimal arrangement of chairs for that number of students. IO can be in any reasonable format. Here is sample output for any number of students from 1 to 100:

```<code>1:  (1, 1)
2:  (1, 2)
3:  (2, 2)
4:  (2, 2)
5:  (2, 3)
6:  (2, 3)
7:  (3, 3)
8:  (3, 3)
9:  (3, 3)
10: (2, 5)
11: (3, 4)
12: (3, 4)
13: (4, 4)
14: (4, 4)
15: (4, 4)
16: (4, 4)
17: (3, 6)
18: (3, 6)
19: (4, 5)
20: (4, 5)
21: (3, 7)
22: (5, 5)
23: (5, 5)
24: (5, 5)
25: (5, 5)
26: (4, 7)
27: (4, 7)
28: (4, 7)
29: (5, 6)
30: (5, 6)
31: (4, 8)
32: (4, 8)
33: (6, 6)
34: (6, 6)
35: (6, 6)
36: (6, 6)
37: (5, 8)
38: (5, 8)
39: (5, 8)
40: (5, 8)
41: (6, 7)
42: (6, 7)
43: (5, 9)
44: (5, 9)
45: (5, 9)
46: (7, 7)
47: (7, 7)
48: (7, 7)
49: (7, 7)
50: (5, 10)
51: (6, 9)
52: (6, 9)
53: (6, 9)
54: (6, 9)
55: (7, 8)
56: (7, 8)
57: (6, 10)
58: (6, 10)
59: (6, 10)
60: (6, 10)
61: (8, 8)
62: (8, 8)
63: (8, 8)
64: (8, 8)
65: (6, 11)
66: (6, 11)
67: (7, 10)
68: (7, 10)
69: (7, 10)
70: (7, 10)
71: (8, 9)
72: (8, 9)
73: (7, 11)
74: (7, 11)
75: (7, 11)
76: (7, 11)
77: (7, 11)
78: (9, 9)
79: (9, 9)
80: (9, 9)
81: (9, 9)
82: (7, 12)
83: (7, 12)
84: (7, 12)
85: (8, 11)
86: (8, 11)
87: (8, 11)
88: (8, 11)
89: (9, 10)
90: (9, 10)
91: (7, 13)
92: (8, 12)
93: (8, 12)
94: (8, 12)
95: (8, 12)
96: (8, 12)
97: (10, 10)
98: (10, 10)
99: (10, 10)
100: (10, 10)
</code>```

As usual, this is code-golf, so standard loopholes apply, and the winner is the shortest answer in bytes.

答:
Julia, 87

I think this is one step in direction towards finding a magic function for the problem:

```<code>f(i)=(i+n)÷(i+1)|>j->(j*i<n)+j
_=indmin([sqrt(n)<=i?i-f(i)*(1-i):2n for i=1:n])
_,f(_)
</code>```

It only looks at pairs (i, j=(i+n)/(i+1)) or (i, j+1)

Title: How should you arrange your chairs?

Body:
Julia, 87

I think this is one step in direction towards finding a magic function for the problem:

f(i)=(i+n)÷(i+1)|>j->(j*i<n)+j
_=indmin([sqrt(n)<=i?i-f(i)*(1-i):2n for i=1:n])
_,f(_)

It only looks at pairs (i, j=(i+n)/(i+1)) or (i, j+1)


答:
Jelly, <s>16</s> <s>15</s> 14 bytes

```<code>÷RĊ,Rµạ/+PỤḢịZ
</code>```

Try it online! or verify all test cases.

How it works

```<code>÷RĊ,Rµạ/+PỤḢịZ  Main link. Argument: n

 R              Range; yield [1, ..., n].
÷               Divide n by each k in [1, ..., n].
  Ċ             Ceil; round the quotients up to the nearest integer.
    R           Range; yield [1, ..., n].
   ,            Pair; yield A := [[ ⌈n ÷ 1⌉, ..., ⌈n ÷ n⌉ ], [ 1, ..., n ]].
     µ          Begin a new, monadic chain. Argument: A
      ạ/        Reduce A by absolute difference.
                This yields [ |⌈n ÷ 1⌉ - 1|, ..., |⌈n ÷ n⌉ - n| ].
         P      Product; reduce A by multiplication.
                This yields [ ⌈n ÷ 1⌉ × 1, ..., ⌈n ÷ n⌉ × n].
       +        Add the results to left and right, element by element. This yields
                [ |⌈n ÷ 1⌉ - 1| + ⌈n ÷ 1⌉ × 1, ..., |⌈n ÷ n⌉ - n| + ⌈n ÷ n⌉ × n ].
          Ụ     Grade up; sort the indices of the list of sums by their values.
           Ḣ    Head; extract the first value, which corresponds to the smallest
                sum. Grading up is stable, so this selects the first index of all
                with the smallest sum in case of a tie. In this event, the first
                index will have the highest absolute difference of all indices
                with the smallest sum, meaning that it has the lowest product and,
                therefore, the lowest number of empty chairs.
             Z  Zip; transpose A's rows and columns.
                This yields [[ ⌈n ÷ 1⌉, 1 ], ..., [ ⌈n ÷ n⌉, n ]].
            ị   Retrieve the pair at that index.
</code>```

Title: How should you arrange your chairs?

Body:
You teach a class of students with interesting preferences for how their chairs are arranged. There are 3 very specific requirements they have for how the chairs are arranged:

They most be arranged in a rectangle, even if it means some chairs go empty.

There must be as few empty chairs as possible.

They must be as "squarey" as possible. Squarey-ness is determined by the distance between the width and the height of the rectangle, lower is better. For example, a rectangle that is 4x7 has a squarey-ness of 3.

To be more specific, the "score" of an arrangement is the distance between the width and height plus the number of chairs that would go empty.

Let's take an example. Let's say you have 13 students. You could arrange the chairs any of these ways:

```<code>1x13
2x7
3x5
4x4
</code>```

1x13 is not very squarey. In fact, 1 and 13 are 12 apart, so we give this arrangement 12 points. It also has 0 empty chairs, so we add 0 points, giving this arrangement a score of 12. Not that great.

2x7 is certainly better. 2 and 7 are only 5 apart, so we give this arrangement 5 points. However, if you actually arranged 2 rows of seven chairs, that would take 14 chairs, meaning one chair would be empty. So we add one point, giving this arrangement a score of 6.

We could also do 3x5. 3 and 5 are 2 apart, so +2 points. It takes 15 chairs, meaning we'd have two extra chairs, so another +2 points, for a score of 4.

Last option, 4x4. 4 and 4 are 0 apart, so we give this +0 points. 4x4 takes 16 chairs, so 3 chairs go empty, for a total score of 3. This is the optimal solution.

In case of a tie, the optimal solution is the one with less empty chairs.

The challenge

You must write a program or function that takes an integer and outputs the optimal arrangement of chairs for that number of students. IO can be in any reasonable format. Here is sample output for any number of students from 1 to 100:

```<code>1:  (1, 1)
2:  (1, 2)
3:  (2, 2)
4:  (2, 2)
5:  (2, 3)
6:  (2, 3)
7:  (3, 3)
8:  (3, 3)
9:  (3, 3)
10: (2, 5)
11: (3, 4)
12: (3, 4)
13: (4, 4)
14: (4, 4)
15: (4, 4)
16: (4, 4)
17: (3, 6)
18: (3, 6)
19: (4, 5)
20: (4, 5)
21: (3, 7)
22: (5, 5)
23: (5, 5)
24: (5, 5)
25: (5, 5)
26: (4, 7)
27: (4, 7)
28: (4, 7)
29: (5, 6)
30: (5, 6)
31: (4, 8)
32: (4, 8)
33: (6, 6)
34: (6, 6)
35: (6, 6)
36: (6, 6)
37: (5, 8)
38: (5, 8)
39: (5, 8)
40: (5, 8)
41: (6, 7)
42: (6, 7)
43: (5, 9)
44: (5, 9)
45: (5, 9)
46: (7, 7)
47: (7, 7)
48: (7, 7)
49: (7, 7)
50: (5, 10)
51: (6, 9)
52: (6, 9)
53: (6, 9)
54: (6, 9)
55: (7, 8)
56: (7, 8)
57: (6, 10)
58: (6, 10)
59: (6, 10)
60: (6, 10)
61: (8, 8)
62: (8, 8)
63: (8, 8)
64: (8, 8)
65: (6, 11)
66: (6, 11)
67: (7, 10)
68: (7, 10)
69: (7, 10)
70: (7, 10)
71: (8, 9)
72: (8, 9)
73: (7, 11)
74: (7, 11)
75: (7, 11)
76: (7, 11)
77: (7, 11)
78: (9, 9)
79: (9, 9)
80: (9, 9)
81: (9, 9)
82: (7, 12)
83: (7, 12)
84: (7, 12)
85: (8, 11)
86: (8, 11)
87: (8, 11)
88: (8, 11)
89: (9, 10)
90: (9, 10)
91: (7, 13)
92: (8, 12)
93: (8, 12)
94: (8, 12)
95: (8, 12)
96: (8, 12)
97: (10, 10)
98: (10, 10)
99: (10, 10)
100: (10, 10)
</code>```

As usual, this is code-golf, so standard loopholes apply, and the winner is the shortest answer in bytes.

Body:
Jelly, <s>16</s> <s>15</s> 14 bytes

÷RĊ,Rµạ/+PỤḢịZ

Try it online! or verify all test cases.

How it works

÷RĊ,Rµạ/+PỤḢịZ  Main link. Argument: n

 R              Range; yield [1, ..., n].
÷               Divide n by each k in [1, ..., n].
  Ċ             Ceil; round the quotients up to the nearest integer.
    R           Range; yield [1, ..., n].
   ,            Pair; yield A := [[ ⌈n ÷ 1⌉, ..., ⌈n ÷ n⌉ ], [ 1, ..., n ]].
     µ          Begin a new, monadic chain. Argument: A
      ạ/        Reduce A by absolute difference.
                This yields [ |⌈n ÷ 1⌉ - 1|, ..., |⌈n ÷ n⌉ - n| ].
         P      Product; reduce A by multiplication.
                This yields [ ⌈n ÷ 1⌉ × 1, ..., ⌈n ÷ n⌉ × n].
       +        Add the results to left and right, element by element. This yields
                [ |⌈n ÷ 1⌉ - 1| + ⌈n ÷ 1⌉ × 1, ..., |⌈n ÷ n⌉ - n| + ⌈n ÷ n⌉ × n ].
          Ụ     Grade up; sort the indices of the list of sums by their values.
           Ḣ    Head; extract the first value, which corresponds to the smallest
                sum. Grading up is stable, so this selects the first index of all
                with the smallest sum in case of a tie. In this event, the first
                index will have the highest absolute difference of all indices
                with the smallest sum, meaning that it has the lowest product and,
                therefore, the lowest number of empty chairs.
             Z  Zip; transpose A's rows and columns.
                This yields [[ ⌈n ÷ 1⌉, 1 ], ..., [ ⌈n ÷ n⌉, n ]].
            ị   Retrieve the pair at that index.


答:
Haskell, 65 bytes

```<code>f x=snd$minimum[((a*b+a-b,a*b),(b,a))|a<-[1..x],b<-[1..a],a*b>=x]
</code>```

Usage example: map f [1..5] -> [(1,1),(1,2),(2,2),(2,2),(2,3)].

Goes through an outer loop a from 1 to x (x -> number of students) and an inner loop b from 1 to a. Keeps all (b,a) where a*b>=x and builds pairs of ((arrangement points,seats left), (b,a)) which follow the lexicographical order we need to find the minimum. Note: a is always greater than b, so we don't need abs for squareyness. No need to subtract x from the "seats left" score, because only the relative order matters. Finally we remove the score pair with snd.

Title: How should you arrange your chairs?

Body:
Haskell, 65 bytes

f x=snd$minimum[((a*b+a-b,a*b),(b,a))|a<-[1..x],b<-[1..a],a*b>=x]

Usage example: map f [1..5] -> [(1,1),(1,2),(2,2),(2,2),(2,3)].

Goes through an outer loop a from 1 to x (x -> number of students) and an inner loop b from 1 to a. Keeps all (b,a) where a*b>=x and builds pairs of ((arrangement points,seats left), (b,a)) which follow the lexicographical order we need to find the minimum. Note: a is always greater than b, so we don't need abs for squareyness. No need to subtract x from the "seats left" score, because only the relative order matters. Finally we remove the score pair with snd.


问:
Recursive Steiner Chains
Steiner Chains are a set of N circles where each circle is tangent to 2 other non-intersecting circles as well as the the previous and next circles of the chain, as seen in the below images:

In this challenge, you will write a program/function that draws Steiner chains recursively, that is, circles of a given chain will be the base circles of another iteration of chains:

Challenge

Write a program/function that accepts image dimensions and a list of integers denoting the level of circles in each successive iteration of chains, and output an image with the recursive Steiner chains drawn to it.

Input

Your program/function will accept 2 arguments:

s - width and height of image

ls - list of positive integers denoting the number of circles present in each successive iteration of chains, ordered from the top-most chain to the bottom-most chain

Output

Your program/function will output an image of dimension s x s displaying the recusive Steiner chain.

The top level base circle will be as large as the image with a diameter of s, centered inside the image

To make things easy, the 2 base circles of a Steiner chain will be concentric, that is, the centerpoints of the 2 baseline circles will be the same

Given an outer radius, R, and the number of circles in a chain, N, the formula for the inner radius R' is R' = (R-R*sin(pi/N))/(sin(pi/N)+1)

Circles of the chain as well as the inner base circle will be the outer base circles of the next iteration of chains

While recursing through the chain circles, the order of the next chain should correspond to the next value in ls

While recursing through the inner circle of a chain, the order should be the same as its parents order (example [5,2]):

All chains should end recursion at a depth of the length of ls

The rotation of the chains doesn't matter:

However, the rotations of recursive chains relative to their parents centerpoint should be the same:

All circles should be drawn with an outline or solid fill

Color choice is left to the implementation, save for loopholes (for example, filling everything with the same color)

Example Runs

In the following examples, color is determined by (depth of the recursion)^4.

You can find source here.

```<code>chain(600,[5,4,3])
</code>```

```<code>chain(600,[11,1,1,1,1,1,1])
</code>```

```<code>chain(600,[5,6,7,8,9])
</code>```

答:
Javascript ES6, 379 bytes

This solution was used to generate the example runs in the question.

```<code>f=(s,ls)=>{with(V=document.createElement`canvas`)with(getContext`2d`)with(Math)return(width=height=s,translate(s/=2,s),(S=(o,d=0,n=ls[d],i=(o-o*sin(PI/n))/(sin(PI/n)+1),r=0)=>{fillStyle=`rgba(0,0,0,${pow(d/ls.length,4)})`;beginPath(),arc(0,0,o,-PI,PI),fill();if(d++<ls.length){S(i,d,n);for(;r<n;++r){save();translate(0,(o+i)/2);S((o-i)/2,d);restore();rotate((2*PI)/n);}}})(s),V)}
</code>```

Ungolfed:

```<code>f=(s,ls)=>{                                        // define function that accepts image dimensions and a list of orders
 with(V=document.createElement`canvas`)            // create canvas to draw on, bring its functions into current scope chain
 with(getContext`2d`)                              // bring graphics functions into current scope chain
 with(Math)return(                                 // bring Math functions into current scope chain
  width=height=s,                                  // set width and height of image
  translate(s/=2,s),                               // center the transform on image
   (S=(o,d=0,                                      // define recursive function that accepts outer radius, depth, and optionally order
       n=ls[d],                                    // default chain order to corresponding order in input list
       i=(o-o*sin(PI/n))/(sin(PI/n)+1),            // calculate inner base circle radius
       r=0)=>{                                     // initialize for loop var
    fillStyle=`rgba(0,0,0,${pow(d/ls.length,4)})`; // fill based on depth
    beginPath(),arc(0,0,o,-PI,PI),fill();          // draw circle
    if(d++<ls.length){                             // if within recursion limit
     S(i,d,n);                                     //   recurse on inner circle
     for(;r<n;++r){                                //   loop through all circles of the chain
      save();                                      //   save transform
      translate(0,(o+i)/2);                        //   translate origin to middle of the 2 base circles
      S((o-i)/2,d);                                //   recurse on chain circle
      restore();                                   //   restore transform
      rotate((2*PI)/n);                            //   rotate transform to next circle in chain
   }}})(s),                                        // begin the recursion
 V)}                                               // return the canvas
</code>```

Note: f returns a canvas.

Example run (assumes there's a  to append to):

```<code>document.body.appendChild(f(600,[13,7,11,5,3]))
</code>```

Should dump the following image to the page:

Title: Recursive Steiner Chains

Body:
Javascript ES6, 379 bytes

This solution was used to generate the example runs in the question.

f=(s,ls)=>{with(V=document.createElement`canvas`)with(getContext`2d`)with(Math)return(width=height=s,translate(s/=2,s),(S=(o,d=0,n=ls[d],i=(o-o*sin(PI/n))/(sin(PI/n)+1),r=0)=>{fillStyle=`rgba(0,0,0,${pow(d/ls.length,4)})`;beginPath(),arc(0,0,o,-PI,PI),fill();if(d++<ls.length){S(i,d,n);for(;r<n;++r){save();translate(0,(o+i)/2);S((o-i)/2,d);restore();rotate((2*PI)/n);}}})(s),V)}

Ungolfed:

f=(s,ls)=>{                                        // define function that accepts image dimensions and a list of orders
 with(V=document.createElement`canvas`)            // create canvas to draw on, bring its functions into current scope chain
 with(getContext`2d`)                              // bring graphics functions into current scope chain
 with(Math)return(                                 // bring Math functions into current scope chain
  width=height=s,                                  // set width and height of image
  translate(s/=2,s),                               // center the transform on image
   (S=(o,d=0,                                      // define recursive function that accepts outer radius, depth, and optionally order
       n=ls[d],                                    // default chain order to corresponding order in input list
       i=(o-o*sin(PI/n))/(sin(PI/n)+1),            // calculate inner base circle radius
       r=0)=>{                                     // initialize for loop var
    fillStyle=`rgba(0,0,0,${pow(d/ls.length,4)})`; // fill based on depth
    beginPath(),arc(0,0,o,-PI,PI),fill();          // draw circle
    if(d++<ls.length){                             // if within recursion limit
     S(i,d,n);                                     //   recurse on inner circle
     for(;r<n;++r){                                //   loop through all circles of the chain
      save();                                      //   save transform
      translate(0,(o+i)/2);                        //   translate origin to middle of the 2 base circles
      S((o-i)/2,d);                                //   recurse on chain circle
      restore();                                   //   restore transform
      rotate((2*PI)/n);                            //   rotate transform to next circle in chain
   }}})(s),                                        // begin the recursion
 V)}                                               // return the canvas

Note: f returns a canvas.

Example run (assumes there's a <body> to append to):

document.body.appendChild(f(600,[13,7,11,5,3]))

Should dump the following image to the page: