A268060 -id:A268060 - cp's OEIS frontend

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.

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 *)

A268058 Maximum value of n-th row of A268057.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 3, 3, 3, 3, 5, 3, 4, 4, 3, 3, 5, 4, 6, 4, 4, 5, 6, 3, 5, 4, 5, 4, 5, 4, 6, 4, 5, 6, 7, 4, 5, 6, 5, 4, 6, 4, 6, 5, 5, 7, 8, 4, 7, 5, 6, 5, 9, 5, 6, 5, 6, 6, 8, 4, 7, 6, 6, 5, 7, 5, 8, 7, 7, 7, 6, 4, 6, 6, 5, 7, 7, 6, 8, 5, 5, 6, 9, 5, 6, 6, 7

Offset: 1

Peter Kagey, Jan 25 2016

nonn

Peter Kagey, Table of n, a(n) for n = 1..10000
Zachary Chase and Mayank Pandey, On the length of Pierce expansions, arXiv preprint (2022). arXiv:2211.08374 [math.NT]
P. Erdős and J. O. Shallit, New bounds on the length of finite Pierce and Engel series, Journal de Théorie des Nombres de Bordeaux 3:1 (1991), pp. 43-53.
Vlado Kešelj, Length of finite Pierce series: theoretical analysis and numerical calculations (1996), 27 pp.
J. O. Shallit, Metric theory of Pierce expansions, Fibonacci Quart. 24 (1986), pp. 22-40.
Reddit user zifyoip, First 100 terms.
Index entries for sequences related to Engel expansions

Cf. A268057, A268059, A268060.

P(a,b)=my(n); while(b, b=a%b; n++); n
a(n)=my(t=1); for(b=2,n-1, t=max(P(n,b),t)); t \\ Charles R Greathouse IV, Nov 26 2016

Chase & Pandey prove that a(n) = O(n^e) for any e > 19/59 = 0.322..., improving on Kešelj, Erdős & Shallit, and Shallit. - Charles R Greathouse IV, Jan 13 2023

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

Original entry on oeis.org

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.

cp's OEIS Frontend

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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A268058 Maximum value of n-th row of A268057.

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

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