A268058 -id:A268058 - cp's OEIS frontend

A268059 a(n) is the number of k such that A268057(n, k) = A268058(n).

1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 4, 4, 2, 1, 1, 1, 1, 1, 2, 5, 1, 4, 2, 3, 5, 2, 1, 3, 2, 1, 1, 2, 5, 1, 4, 3, 1, 5, 2, 1, 1, 1, 1, 3, 1, 2, 1, 3, 2, 4, 1, 1, 2, 3, 2, 6, 2, 3, 1, 2, 1, 2, 1, 1, 1, 1, 6, 7, 5, 2, 7, 1, 5, 1, 2, 3, 10, 3, 2, 2, 2, 4, 2

Offset: 1

Peter Kagey, Jan 25 2016

nonn

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A268057, A268058, A268060.

A268060 Least k such that A268057(n, k) = A268058(n).

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 8, 9, 10, 11, 12, 13, 13, 14, 14, 13, 16, 15, 16, 17, 16, 17, 20, 19, 20, 25, 22, 22, 21, 24, 22, 23, 24, 23, 27, 28, 29, 29, 30, 29, 31, 32, 31, 33, 32, 32, 36, 33, 35, 35, 34, 34, 37, 39, 38, 41, 41, 40, 43, 43, 44

Offset: 1

Peter Kagey, Jan 25 2016

nonn

a(n) > n/2 for all n > 2.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A268057, A268058, A268059.

A268057 Triangle T(n,k), 1<=k<=n, read by rows: T(n,k) = number of iterations of A048158(n, A048158(n, ... A048158(n, k)...)) to reach 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 3, 2, 1, 1, 1, 2, 1, 3, 2, 2, 1, 1, 2, 1, 2, 3, 2, 3, 2, 1, 1, 1, 2, 2, 1, 3, 3, 2, 2, 1, 1, 2, 3, 4, 2, 3, 5, 4, 3, 2, 1, 1, 1, 1, 1, 2, 1, 3, 2, 2, 2, 2, 1, 1, 2, 2, 2, 3, 2, 3, 4, 3

Offset: 1

Peter Kagey, Jan 25 2016

Each column is periodic: T(n+A003418(k),k) = T(n,k). - Robert Israel, Feb 02 2016

			T(5, 3) = 3 because the algorithm requires three steps to reach 0.
  5 % 3 = 2
  5 % 2 = 1
  5 % 1 = 0
Triangle begins:
  1
  1 1
  1 2 1
  1 1 2 1
  1 2 3 2 1
  1 1 1 2 2 1
  1 2 2 3 3 2 1
  1 1 2 1 3 2 2 1
  1 2 1 2 3 2 3 2 1
  1 1 2 2 1 3 3 2 2 1
  1 2 3 4 2 3 5 4 3 2 1
  1 1 1 1 2 1 3 2 2 2 2 1

Peter Kagey, Table of n, a(n) for n = 1..10000
"ModernModest", Reddit discussion

Cf. A003418, A048158, A107435, A268058, A268059, A268060.

T:= proc(n,k) option remember; local m;
     if k = 0 then 0 else 1 + procname(n,n mod k) fi
end proc:
seq(seq(T(n,k),k=1..n),n=1..30); # Robert Israel, Feb 02 2016

T[n_, k_] := T[n, k] = If[k == 0, 0, 1 + T[n, Mod[n, k]]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 30}] // Flatten (* Jean-François Alcover, Jan 31 2023, after Robert Israel *)

A006538 Worst cases for Pierce expansions (denominators).

Original entry on oeis.org

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

Jeffrey Shallit, N. J. A. Sloane

See A006537 for numerators.

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

RECORDS transform of A268058.

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

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

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

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A268059 a(n) is the number of k such that A268057(n, k) = A268058(n).

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A268060 Least k such that A268057(n, k) = A268058(n).

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A268057 Triangle T(n,k), 1<=k<=n, read by rows: T(n,k) = number of iterations of A048158(n, A048158(n, ... A048158(n, k)...)) to reach 0.

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A006538 Worst cases for Pierce expansions (denominators).

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