A369162 a(n) = A000688(A000688(n)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Jan 15 2024

Keywords

• nonn
• easy

Comments

First differs from A364388 at n = 42.

The sums of the first 10^k terms, for k = 1, 2, ..., are 10, 102, 1024, 10285, 102988, 1030280, 10304021, 103043644, 1030448091, 10304515936, ... . From these values the asymptotic mean of this sequence, whose existence was proven by Ivić (1983) (see the Formula section), can be empirically evaluated by 1.0304... .

References

• József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter XIII, page 477-478.

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, Bull. Cl. Sci. Math. Nat., Sci. Math., Vol. 17 (1989), pp. 13-22; alternative link.
• Aleksandar Ivić, On the number of abelian groups of a given order and on certain related multiplicative functions, Journal of Number Theory, Vol. 16, No. 1 (1983), pp. 119-137. See p. 131, eq. 4.2.

Crossrefs

Cf. A000688, A364388.

Cf. A048109, A059956, A271971.

Cf. A369163, A369164, A369165, A369167.

Programs

Mathematica

Table[FiniteAbelianGroupCount[FiniteAbelianGroupCount[n]], {n, 1, 100}]

PARI

A000688(n) = vecprod(apply(numbpart, factor(n)[, 2]));
a(n) = A000688(A000688(n));

Formula

Sum_{k=1..n} a(k) = c * n + O(sqrt(n) * log(n)^4), where c = Sum_{k>=1} d(k) * A000688(k) is a constant, d(k) is the asymptotic density of the set {m | A000688(m) = k} (e.g., d(1) = A059956, d(2) = A271971, d(3) appears in A048109) (Ivić, 1983).