The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A296523 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x > 0, y >= 0 and 0 <= z <= w such that 64*x^2 + 65*y^2 = q^2 for some practical number q. 1
 1, 1, 1, 1, 3, 3, 1, 1, 4, 3, 2, 1, 3, 4, 1, 1, 4, 4, 2, 3, 5, 4, 1, 3, 5, 5, 3, 1, 6, 6, 1, 1, 5, 5, 3, 4, 4, 6, 1, 3, 8, 4, 2, 2, 9, 6, 1, 1, 6, 8, 4, 3, 6, 10, 3, 4, 6, 4, 5, 1, 7, 7, 2, 1, 9, 8, 2, 5, 9, 10 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Conjecture: (i) a(n) > 0 for all n > 0. Also, a(n) = 1 only for those numbers 2^k (k = 0,1,2,...), 4^k*m (k = 0,1,...; m = 3, 7), 4^k*47 (k = 0,1,2,3), 4^k*s (k = 0,1,2; s = 15, 23, 31, 39, 71); 4^k*t (k =0,1; t = 87, 111, 119, 159, 191, 311). (ii) Any positive integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that a*x^2 + b*y^2 = q^2 for some practical number q, provided that (a,b) is among the ordered pairs (5,16), (7,36), (16,77), (36,55), (36,91), (36, -5), (64,-7). LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190. Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017. Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT].) EXAMPLE a(3) = 1 since 3 = 1^2 + 0^2 + 1^2 + 1^2 and 64*1^2 + 65*0^2 = 8^2 with 8 practical. a(71) = 1 since 71 = 3^2 + 6^2 + 1^2 + 5^2 and 64*3^2 + 65*6^2 = 54^2 with 54 practical. a(159) = 1 since 159 = 5^2 + 10^2 + 3^2 + 5^2 and 64*5^2 + 65*10^2 = 90^2 with 90 practical. a(311) = 1 since 311 = 1^2 + 2^2 + 9^2 + 15^2 and 64*1^2 + 65*2^2 = 18^2 with 18 practical. MATHEMATICA SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]; f[n_]:=f[n]=FactorInteger[n]; Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2]); Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}]; pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0); pQ[n_]:=pQ[n]=SQ[n]&&pr[Sqrt[n]]; tab={}; Do[r=0; Do[If[SQ[n-x^2-y^2-z^2]&&pQ[64x^2+65y^2], r=r+1], {x, 1, Sqrt[n]}, {y, 0, Sqrt[n-x^2]}, {z, 0, Sqrt[(n-x^2-y^2)/2]}]; tab=Append[tab, r], {n, 1, 70}] CROSSREFS Cf. A000118, A000290, A271518, A281976. Sequence in context: A109439 A247646 A133333 * A171876 A306462 A133332 Adjacent sequences:  A296520 A296521 A296522 * A296524 A296525 A296526 KEYWORD nonn AUTHOR Zhi-Wei Sun, Dec 14 2017 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 16 12:38 EDT 2021. Contains 343947 sequences. (Running on oeis4.)
{"ft_lang_label":"__label__en","ft_lang_prob":0.5614251,"math_prob":0.99966574,"size":2688,"snap":"2021-21-2021-25","text_gpt3_token_len":1368,"char_repetition_ratio":0.114754096,"word_repetition_ratio":0.027777778,"special_character_ratio":0.6063988,"punctuation_ratio":0.2816568,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9996991,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-16T16:41:36Z\",\"WARC-Record-ID\":\"<urn:uuid:25b55bb7-35d1-49f8-b55d-46654d1f9acc>\",\"Content-Length\":\"19107\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6de7b684-059d-440b-8d59-b66a7af9d574>\",\"WARC-Concurrent-To\":\"<urn:uuid:6ca690af-f8ab-4a25-b159-108906827147>\",\"WARC-IP-Address\":\"104.239.138.29\",\"WARC-Target-URI\":\"https://oeis.org/A296523\",\"WARC-Payload-Digest\":\"sha1:HK4VKKW2GV7JAKVC6BAL3R2PHRF7SO6Z\",\"WARC-Block-Digest\":\"sha1:DGWPJDBN66WLVTUJ4Q4WY5THHGS7ZPOD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243991224.58_warc_CC-MAIN-20210516140441-20210516170441-00319.warc.gz\"}"}