A369167 a(n) = A000688(n + A000688(n)).

1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 5, 1, 2, 2, 2, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 1, 7, 1, 1, 1, 4, 3, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 3, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3

Offset: 1

Author

Amiram Eldar, Jan 15 2024

Keywords

• nonn
• easy

References

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Haihong Fan and Wenguang Zhai, A Symmetric Form of the Mean Value Involving Non-Isomorphic Abelian Groups, Symmetry 2022, 14(9), 1755.
• Haihong Fan and Wenguang Zhai, On some sums involving the counting function of nonisomorphic Abelian groups, Lithuanian Mathematical Journal, Vol. 63 (2023), pp. 166-180; arXiv preprint, arXiv:2204.02576 [math.NT], 2022.
• Aleksandar Ivić, An asymptotic formula involving the enumerating function of finite abelian groups, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 3 (1992), pp. 61-66.

Crossrefs

Cf. A000688, A369162.

Programs

Mathematica

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

PARI

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

Formula

Sum_{k=1..n} a(k) = c * n + O(n^(k+eps)) for any eps > 0, where c > 0 is a constant and k = 11/12 (Ivić, 1992), 3/4 (Fan and Zhai, 2023), or 2/3 (Fan and Zhai, 2022).