A328705 Dirichlet g.f.: Product_{k>=1} zeta(k*s)^2.

1, 2, 2, 5, 2, 4, 2, 10, 5, 4, 2, 10, 2, 4, 4, 20, 2, 10, 2, 10, 4, 4, 2, 20, 5, 4, 10, 10, 2, 8, 2, 36, 4, 4, 4, 25, 2, 4, 4, 20, 2, 8, 2, 10, 10, 4, 2, 40, 5, 10, 4, 10, 2, 20, 4, 20, 4, 4, 2, 20, 2, 4, 10, 65, 4, 8, 2, 10, 4, 8, 2, 50, 2, 4, 10, 10, 4, 8, 2, 40

Offset: 1

Author

Ilya Gutkovskiy, Oct 26 2019

Keywords

• nonn
• mult

Comments

Dirichlet convolution of A000688 with itself.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000688, A000712, A001620, A063966, A129667, A188581, A188585, A301830.

Programs

Mathematica

Table[DivisorSum[n, FiniteAbelianGroupCount[n/#] FiniteAbelianGroupCount[#] &], {n, 1, 80}]

Formula

a(n) = Sum_{d|n} A000688(n/d) * A000688(d).

Sum_{k=1..n} a(k) ~ c^2 * n * (log(n) + 2*gamma - 1 - 2*s), where c = A021002 = Product_{k>=2} zeta(k) = 2.2948565916733137941835158313443112887131637994..., s = Sum_{k>=2} k*zeta'(k)/zeta(k) = -2.1955691982567064617939038695473479681910375... and gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 26 2019

Multiplicative with a(p^e) = A000712(e). - Amiram Eldar, Nov 30 2020