A340850 Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s) * zeta(s-2) / (zeta(s-1))^2.

1, 1, 4, 5, 16, 4, 36, 21, 40, 16, 100, 20, 144, 36, 64, 85, 256, 40, 324, 80, 144, 100, 484, 84, 416, 144, 364, 180, 784, 64, 900, 341, 400, 256, 576, 200, 1296, 324, 576, 336, 1600, 144, 1764, 500, 640, 484, 2116, 340, 1800, 416, 1024, 720, 2704, 364, 1600, 756, 1296, 784

Offset

1,3

Links

Aloe Poliszuk, Table of n, a(n) for n = 1..10000

Formula

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

Dirichlet convolution of A002618 and A023900.

Dirichlet convolution of A001157 and A328722.

Dirichlet inverse b(n) for n > 0 is multiplicative with b(1) = 1 and b(p^e) = -(p-1)^2 * e * p^(e-1) for prime p and e > 0.

Dirichlet convolution with A060640 equals A007433.

Dirichlet convolution with A018804 equals A000290.

Sum_{k=1..n} a(k) ~ c * n^3, where c = 12*zeta(3)/Pi^4 = 0.148083... . - Amiram Eldar, Oct 16 2022

a(n) = A000203(n^2 / rad(n)) * A173557(n)^2 / A048250(n). - Aloe Poliszuk, Nov 15 2025

Mathematica

f[p_, e_] := (p^(2*e) - 1)*(p - 1)/(p + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 24 2021 *)

Prog

(PARI) a(n)=my(f=factorint(n)); prod(i=1, #f~, (f[i, 1]^(2*f[i, 2])-1)*(f[i, 1]-1)/(f[i, 1]+1)); \\ Aloe Poliszuk, Nov 15 2025

Crossrefs

Cf. A000203, A000290, A001157, A002618, A007433, A018804, A023900, A048250, A060640, A173557, A328722.

Sequence in context: A289021 A323627 A289742 * A249113 A166590 A368739

Adjacent sequences: A340847 A340848 A340849 * A340851 A340852 A340853

Keyword

nonn,easy,mult

Author

Werner Schulte, Jan 24 2021

Status

approved