A369911 a(n) = Sum_{p|n, p prime} p * sopf(n/p).

0, 0, 0, 4, 0, 12, 0, 4, 9, 20, 0, 16, 0, 28, 30, 4, 0, 21, 0, 24, 42, 44, 0, 16, 25, 52, 9, 32, 0, 62, 0, 4, 66, 68, 70, 25, 0, 76, 78, 24, 0, 82, 0, 48, 39, 92, 0, 16, 49, 45, 102, 56, 0, 21, 110, 32, 114, 116, 0, 66, 0, 124, 51, 4, 130, 122, 0, 72, 138, 118, 0, 25

Offset: 1

Author

Wesley Ivan Hurt, Feb 05 2024

Keywords

• nonn
• easy

Comments

Dirichlet convolution of A061397(n) and A008472(n). - Wesley Ivan Hurt, Jul 10 2025

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A008472 (sopf), A061397, A369744.

Programs

Mathematica

a[n_] := Sum[p, {p, Select[Divisors[n], PrimeQ]}]; Table[DivisorSum[n, #*a[n/#] &, PrimeQ[#] &], {n, 100}]

PARI

A008472(n) = vecsum(factor(n)[, 1]);
A369911(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i,1]*A008472(n/f[i, 1]))); \\ Antti Karttunen, Jan 23 2025

Formula

From Wesley Ivan Hurt, Jul 10 2025: (Start)

a(n) = Sum_{d|n} A061397(d) * A008472(n/d).

a(p^k) = p^2 * (1-floor(1/k)) for p prime and k>=1. (End)