A006538 Worst cases for Pierce expansions (denominators). (Formerly M2471)

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

1,2

Comments

See A006537 for numerators.

a(58) <= 58017959. - Hiroaki Yamanouchi, Aug 31 2014

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

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

Prog

(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

Crossrefs

RECORDS transform of A268058.

Sequence in context: A065019 A071328 A273037 * A066281 A072063 A242269

Adjacent sequences: A006535 A006536 A006537 * A006539 A006540 A006541

Keyword

nonn,frac

Author

Jeffrey Shallit, N. J. A. Sloane

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

Status

approved