A135063 Define the sequence {b_n(m)} by b_n(0)=0; 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, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 4, 1, 3, 1, 3, 1, 2, 2, 1, 4, 1, 1, 2, 3, 1, 1, 1, 1, 2, 1, 2, 3, 1, 2, 2, 5, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 4, 3, 1, 1, 1, 1, 2, 1, 1

Offset

1,2

Links

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Example

{b_7(m)} is 0,2,3,4,2,3,4,..., with (2,3,4) repeating thereafter. So a(7) = 3, the length of the repeating subsequence (2,3,4).

Maple

b := proc(n, m) option remember; if m =0 then 0; else numtheory[tau]( procname(n, m-1)+n); end if; end proc:
A135063 := proc(n) bseq := [] ; for m from 0 do bs := b(n, m) ; if member(bs, bseq, 'w') then return 1+nops(bseq)-w ; else bseq := [op(bseq), bs] ; end if; end do: end proc: seq(A135063(n), n=1..120) ; # R. J. Mathar, Aug 09 2010

Mathematica

b[n_, m_] := b[n, m] = If[m == 0, 0, DivisorSigma[0, b[n, m-1] + n]];
A135063[n_] := Module[{bseq = {}}, For[m = 0, True, m++, bs = b[n, m]; w = Position[bseq, bs]; If[w != {}, Return[1+Length[bseq]-w[[1, 1]]], AppendTo[bseq, bs]]]];
Table[A135063[n], {n, 1, 120}] (* Jean-François Alcover, Feb 08 2024, after R. J. Mathar *)

Prog

(PARI) a(n) = my(b, k, v=List([0])); 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, A135062.

Sequence in context: A129265 A030358 A118914 * A370770 A124010 A212171

Adjacent sequences: A135060 A135061 A135062 * A135064 A135065 A135066

Keyword

nonn

Author

Leroy Quet, Nov 15 2007

Extensions

Terms beyond a(12) from R. J. Mathar, Aug 09 2010

Status

approved