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A006538 Worst cases for Pierce expansions (denominators).

%I A006538 M2471 #32 Jan 05 2025 19:51:34
%S A006538 1,3,5,11,11,19,35,47,53,95,103,179,251,299,503,743,1019,1319,1439,
%T A006538 2939,3359,3959,5387,5387,5879,5879,17747,17747,23399,23399,23399,
%U A006538 23399,23399,23399,93596,186479,186479,278387,442679,493919,493919,493919,830939,1371719,1371719,1371719,1371719,1371719,1371719
%N A006538 Worst cases for Pierce expansions (denominators).
%C A006538 See A006537 for numerators.
%C A006538 a(58) <= 58017959. - _Hiroaki Yamanouchi_, Aug 31 2014
%D A006538 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006538 Hiroaki Yamanouchi, <a href="/A006538/b006538.txt">Table of n, a(n) for n = 1..57</a>
%H A006538 P. Erdős and Jeffrey Shallit, <a href="http://www.numdam.org/item?id=JTNB_1991__3_1_43_0">New bounds on the length of finite Pierce and Engel series</a>, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
%H A006538 Vlado Kešelj, <a href="http://www.cs.uwaterloo.ca/research/tr/1996/21/cs-96-21.pdf">Length of finite Pierce series: theoretical analysis and numerical computations</a>, Dept. Computer Science, U Waterloo, CS-96-21, Sep 10 1996.
%H A006538 M. E. Mays, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/25-3/mays.pdf">Iterating the division algorithm</a>, Fib. Quart., 25 (1987), 204-213.
%H A006538 J. O. Shallit, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/24-1/shallit.pdf">Metric theory of Pierce expansions</a>, Fibonacci Quart. 24 (1986), pp. 22-40.
%H A006538 <a href="/index/El#Engel">Index entries for sequences related to Engel expansions</a>
%F A006538 Chase & Pandey prove that a(n) >> n^e for some e > 59/19 = 3.105..., improving on Kešelj, Erdős & Shallit, and Shallit. - _Charles R Greathouse IV_, Jan 14 2023
%o A006538 (PARI) P(a, b)=my(n); while(b, b=a%b; n++); n
%o A006538 A268058(n)=my(t=1); for(b=2, n-1, t=max(P(n, b), t)); t
%o A006538 a(n,startAt=1)=while(A268058(startAt) < n, startAt++); startAt \\ _Charles R Greathouse IV_, Jan 14 2023
%Y A006538 RECORDS transform of A268058.
%K A006538 nonn,frac
%O A006538 1,2
%A A006538 _Jeffrey Shallit_, _N. J. A. Sloane_
%E A006538 Description corrected May 15 1995 and again Nov 07 2006
%E A006538 a(38)-a(49) (from Keselj report) added by _R. J. Mathar_, Jun 30 2008