A329354 a(n) = Sum_{d|n} d*omega(d).

0, 2, 3, 6, 5, 17, 7, 14, 12, 27, 11, 45, 13, 37, 38, 30, 17, 62, 19, 71, 52, 57, 23, 101, 30, 67, 39, 97, 29, 162, 31, 62, 80, 87, 82, 162, 37, 97, 94, 159, 41, 220, 43, 149, 137, 117, 47, 213, 56, 152, 122, 175, 53, 197, 126, 217, 136, 147, 59, 410, 61, 157, 187, 126, 148, 336, 67, 227, 164, 342, 71, 362, 73, 187, 213, 253, 172, 394, 79

Offset: 1

Author

Antti Karttunen, Nov 15 2019

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A001221, A013939, A062799, A180253, A323599, A328260, A329375.

Programs

Mathematica

Table[Sum[d*PrimeNu[d], {d, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Aug 18 2021 *)

PARI

A329354(n) = sumdiv(n,d,omega(d)*d);

Formula

a(n) = Sum_{d|n} d*A001221(d).

a(n) = A180253(n) - A323599(n).

a(n) = A328260(n) + A329375(n).

a(n) = Sum_{d|n} (n/d) * sopf(d). - Wesley Ivan Hurt, May 24 2021

Dirichlet g.f.: primezeta(s-1) * zeta(s-1) * zeta(s). - Ilya Gutkovskiy, Aug 18 2021

Conjecture: Sum_{k=1..n} a(k) ~ Pi^2 * n^2 * (log(log(n)) + A077761) / 12. - Vaclav Kotesovec, Mar 03 2023