A135062 Define the sequence {b_n(m)} by b_n(0)=1; b_n(m) = A000005(b_n(m-1) + n), for all m >= 1. Then a(n) is the smallest positive integer such that b_n(m) = b_n(m + a(n)) for all m > some positive integer.

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

Offset

0,3

Links

Table of n, a(n) for n=0..86.

Example

{b_8(m)} is 1, 3, 2, 4, 6, 4, 6, ..., with (4, 6) repeating thereafter. So a(8) = 2, the length of the repeating subsequence (4, 6).

Mathematica

nmax = 100; mmax = 10; (* sequence shows zeroes if mmax insufficient *)
b[n_, m_] := b[n, m] = If[m == 0, 1, DivisorSigma[0, b[n, m-1] + n]];
perLen[lst_] := FindTransientRepeat[lst, 2] // Last // Length;
a[n_] := Table[b[n, m], {m, 0, mmax}] // perLen;
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Feb 08 2024 *)

Prog

(PARI) a(n) = my(b=1, k, v=List([1])); until(k<#v, k=1; listput(v, b=numdiv(b+n)); until(v[k]==b||k==#v, k++)); #v-k; \\ Jinyuan Wang, Aug 22 2021

Crossrefs

Cf. A000005, A135063.

Sequence in context: A275111 A182961 A339814 * A088428 A025838 A285813

Adjacent sequences: A135059 A135060 A135061 * A135063 A135064 A135065

Keyword

nonn

Author

Leroy Quet, Nov 15 2007

Extensions

More terms from Jinyuan Wang, Aug 22 2021

Status

approved