A369744 a(n) = Sum_{p|n, p prime} p * omega(n/p).

0, 0, 0, 2, 0, 5, 0, 2, 3, 7, 0, 7, 0, 9, 8, 2, 0, 8, 0, 9, 10, 13, 0, 7, 5, 15, 3, 11, 0, 20, 0, 2, 14, 19, 12, 10, 0, 21, 16, 9, 0, 24, 0, 15, 11, 25, 0, 7, 7, 12, 20, 17, 0, 8, 16, 11, 22, 31, 0, 22, 0, 33, 13, 2, 18, 32, 0, 21, 26, 28, 0, 10, 0, 39, 13, 23, 18, 36

Offset: 1

Author

Wesley Ivan Hurt, Jan 30 2024

Keywords

• nonn
• easy

Comments

Dirichlet convolution of A061397(n) and A001221(n). - Wesley Ivan Hurt, Apr 24 2025

Links

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

Crossrefs

Cf. A001221 (omega), A010051, A061397, A323599, A345354, A345355.

Cf. also A369911.

Programs

Mathematica

Table[DivisorSum[n, #*PrimeNu[n/#] &, PrimeQ[#] &], {n, 100}]

PARI

A369744(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i,1]*omega(n/f[i, 1]))); \\ Antti Karttunen, Jan 23 2025

Formula

a(p^k) = 1 for p prime and k = 1, else p if k >= 2. - Wesley Ivan Hurt, Jun 26 2024

a(n) = Sum_{d|n} d * omega(n/d) * c(d), where c = A010051. - Wesley Ivan Hurt, Apr 15 2025