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

1,3

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.

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).

Mathematica

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

Prog

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

Crossrefs

Cf. A000688, A369162.

Sequence in context: A204988 A368332 A224765 * A160267 A115621 A326514

Adjacent sequences: A369164 A369165 A369166 * A369168 A369169 A369170

Keyword

nonn,easy

Author

Amiram Eldar, Jan 15 2024

Status

approved