A006538 Worst cases for Pierce expansions (denominators).
1, 3, 5, 11, 11, 19, 35, 47, 53, 95, 103, 179, 251, 299, 503, 743, 1019, 1319, 1439, 2939, 3359, 3959, 5387, 5387, 5879, 5879, 17747, 17747, 23399, 23399, 23399, 23399, 23399, 23399, 93596, 186479, 186479, 278387, 442679, 493919, 493919, 493919, 830939, 1371719, 1371719, 1371719, 1371719, 1371719, 1371719
Offset: 1
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Hiroaki Yamanouchi, Table of n, a(n) for n = 1..57
- P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
- Vlado Kešelj, Length of finite Pierce series: theoretical analysis and numerical computations, Dept. Computer Science, U Waterloo, CS-96-21, Sep 10 1996.
- M. E. Mays, Iterating the division algorithm, Fib. Quart., 25 (1987), 204-213.
- J. O. Shallit, Metric theory of Pierce expansions, Fibonacci Quart. 24 (1986), pp. 22-40.
- Index entries for sequences related to Engel expansions
Crossrefs
RECORDS transform of A268058.
Programs
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PARI
P(a, b)=my(n); while(b, b=a%b; n++); n A268058(n)=my(t=1); for(b=2, n-1, t=max(P(n, b), t)); t a(n,startAt=1)=while(A268058(startAt) < n, startAt++); startAt \\ Charles R Greathouse IV, Jan 14 2023
Formula
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
Extensions
Description corrected May 15 1995 and again Nov 07 2006
a(38)-a(49) (from Keselj report) added by R. J. Mathar, Jun 30 2008
Comments