A349469 Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s-1)*zeta(s-3)/(zeta(s-2))^2.

1, 2, 12, 20, 80, 24, 252, 168, 360, 160, 1100, 240, 1872, 504, 960, 1360, 4352, 720, 6156, 1600, 3024, 2200, 11132, 2016, 10400, 3744, 9828, 5040, 22736, 1920, 27900, 10912, 13200, 8704, 20160, 7200, 47952, 12312, 22464, 13440, 65600, 6048, 75852, 22000, 28800, 22264, 99452, 16320, 88200, 20800

Offset: 1

Author

Werner Schulte, Nov 18 2021

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A057660, A068963, A340850.

Programs

Mathematica

f[p_, e_] := (p - 1)*p^e*(p^(2*e) - 1)/(p + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 18 2021 *)

Formula

Multiplicative with a(p^e) = p^e * (p^(2*e)-1) * (p-1) / (p+1) for e > 0 and prime p.

Dirichlet convolution with A057660 equals A068963.

Equals n * A340850(n) for n > 0.

Dirichlet inverse b(n) for n > 0 is multiplicative with b(1) = 1 and

b(p^e) = -(p-1)^2 * e * p^(2*e-1) for prime p and e > 0.

Sum_{k=1..n} a(k) ~ c * n^4, where c = 9*zeta(3)/Pi^4 = 0.111062... . - Amiram Eldar, Oct 16 2022