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%I #13 Oct 15 2023 05:12:24
%S 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,
%T 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,
%U 2,2,2,3,2,2,3,3,2,2,2,3,3,2,2,3,2,2,2
%N Number of iterations of A000688 needed to reach 1 starting at n (n is counted).
%C The least value of n such that a(n) = 1, 2, ..., 5 is 1, 2, 4, 36 and 264600.
%H Amiram Eldar, <a href="/A346213/b346213.txt">Table of n, a(n) for n = 1..10000</a>
%H Paul Erdős and Aleksandar Ivić, <a href="https://www.jstor.org/stable/44095772">On the iterates of the enumerating function of finite abelian groups</a>, Bulletin Académie serbe des sciences et des arts, Classe des sciences mathématiques et naturelles, Sciences mathématiques, No. 17 (1989), pp. 13-22; <a href="https://www.renyi.hu/~p_erdos/1989-23.pdf">alternative link</a>.
%F Sum_{k<=x} a(k) ~ c*x + O(x^(1/2 + eps)), where c > 1 is a constant (Erdős and Ivić, 1989).
%e a(4) = 3 since the trajectory of n = 4, {n, A000688(n), A000688(A000688(n))} = {4, 2, 1}, has the length 3.
%t a[n_] := -1 + Length @ FixedPointList[FiniteAbelianGroupCount, n]; Array[a, 100]
%Y Cf. A000688, A060937.
%K nonn
%O 1,2
%A _Amiram Eldar_, Jul 10 2021
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