A346213 Number of iterations of A000688 needed to reach 1 starting at n (n is counted).

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

Offset

1,2

Comments

The least value of n such that a(n) = 1, 2, ..., 5 is 1, 2, 4, 36 and 264600.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Paul Erdős and Aleksandar Ivić, On the iterates of the enumerating function of finite abelian groups, Bulletin Académie serbe des sciences et des arts, Classe des sciences mathématiques et naturelles, Sciences mathématiques, No. 17 (1989), pp. 13-22; alternative link.

Formula

Sum_{k<=x} a(k) ~ c*x + O(x^(1/2 + eps)), where c > 1 is a constant (Erdős and Ivić, 1989).

Example

a(4) = 3 since the trajectory of n = 4, {n, A000688(n), A000688(A000688(n))} = {4, 2, 1}, has the length 3.

Mathematica

a[n_] := -1 + Length @ FixedPointList[FiniteAbelianGroupCount, n]; Array[a, 100]

Crossrefs

Cf. A000688, A060937.

Sequence in context: A104517 A098397 A278744 * A082091 A375431 A334216

Adjacent sequences: A346210 A346211 A346212 * A346214 A346215 A346216

Keyword

nonn

Author

Amiram Eldar, Jul 10 2021

Status

approved