The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A226051 Denominators of signed Egyptian fractions with sums converging to Pi. 2

%I #12 Apr 09 2021 19:04:09

%S 13,25,685,1046508,3249663242463,11242879629599233747822858,

%T 296966610697839275946742496484126789076162611540968

%N Denominators of signed Egyptian fractions with sums converging to Pi.

%C Using the algorithm defined at A226049 with r = Pi and f(n) = 1/n gives r = (Sum_{k=1..13} 1/k) - 1/25 + 1/685 - 1/1046508 + ..., of which the 22nd partial sum differs from Pi by less than 10^(-800). For a guide to related sequences, see A226049.

%H Clark Kimberling, <a href="/A226051/b226051.txt">Table of n, a(n) for n = 1..12</a>

%e 1 + 1/2 + ... + 1/12 < Pi < 1 + 1/2 + ... + 1/13, so a(1) = 13.

%e 1 + 1/2 + ... + 1/13 - 1/25 < Pi, so a(2) = 25.

%e 1 + 1/2 + ... + 1/13 - 1/25 + 1/685 > Pi, so a(3) = 685.

%t \$MaxExtraPrecision = Infinity;

%t nn = 10; f[n_] := 1/n; r = Pi; s = 0; b = NestWhile[# + 1 &, 1, ! (s += f[#]) > r &]; u = Sum[f[n], {n, 1, b}]; c = Floor[1/(u - r)]; v = u - 1/c; n = 1; While[n < nn/2, n++; b[n] = Floor[1/(r - v[n - 1])]; u[n] = v[n - 1] + 1/b[n]; c[n] = Floor[1/(u[n] - r)]; v[n] = u[n] - 1/c[n]]; a = Riffle[Table[b[i], {i, 1, nn/2}], Table[c[i], {i, 1, nn/2}]]

%Y Cf. A226049.

%K nonn

%O 1,1

%A _Clark Kimberling_, May 24 2013

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

Last modified September 24 22:14 EDT 2023. Contains 365582 sequences. (Running on oeis4.)
{"ft_lang_label":"__label__en","ft_lang_prob":0.5721054,"math_prob":0.9974731,"size":1211,"snap":"2023-40-2023-50","text_gpt3_token_len":571,"char_repetition_ratio":0.10439105,"word_repetition_ratio":0.0720339,"special_character_ratio":0.65400493,"punctuation_ratio":0.22865854,"nsfw_num_words":1,"has_unicode_error":false,"math_prob_llama3":0.9964401,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-25T02:57:23Z\",\"WARC-Record-ID\":\"<urn:uuid:99b7d207-3136-4814-b13f-ba0ed608e448>\",\"Content-Length\":\"8901\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:52485414-9cbc-434c-8f4a-7f293167fb40>\",\"WARC-Concurrent-To\":\"<urn:uuid:190cd798-6b13-4bbf-9563-005427038c12>\",\"WARC-IP-Address\":\"104.239.138.29\",\"WARC-Target-URI\":\"https://oeis.org/A226051/internal\",\"WARC-Payload-Digest\":\"sha1:TYKVAP53YYMNVCLGTAL3KAZA53V3NECT\",\"WARC-Block-Digest\":\"sha1:MMB5IVP24IJJCGIWUD3NTR4XMS4GN4NQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233506676.95_warc_CC-MAIN-20230925015430-20230925045430-00346.warc.gz\"}"}