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

Author

Amiram Eldar, Jul 10 2021

Keywords

• nonn

Comments

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

Examples

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

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.

Crossrefs

Cf. A000688, A060937.

Programs

Mathematica

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

Formula

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