A345355 a(n) = Sum_{p|n, p prime} p^omega(n/p).

0, 1, 1, 2, 1, 5, 1, 2, 3, 7, 1, 7, 1, 9, 8, 2, 1, 11, 1, 9, 10, 13, 1, 7, 5, 15, 3, 11, 1, 38, 1, 2, 14, 19, 12, 13, 1, 21, 16, 9, 1, 62, 1, 15, 14, 25, 1, 7, 7, 27, 20, 17, 1, 11, 16, 11, 22, 31, 1, 42, 1, 33, 16, 2, 18, 134, 1, 21, 26, 78, 1, 13, 1, 39, 28, 23, 18, 182

Offset: 1

Author

Wesley Ivan Hurt, Jun 15 2021

Keywords

• nonn

Examples

			a(30) = Sum_{p|30} p^omega(30/p) = 2^omega(15) + 3^omega(10) + 5^omega(6) = 2^2 + 3^2 + 5^2 = 38.
		

Links

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

Crossrefs

Cf. A001221 (omega), A010051, A369744.

Cf. also A369741, A369905, A369907.

Programs

Mathematica

Table[Sum[k^PrimeNu[n/k] (PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]

PARI

a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,1]^omega(n/f[k,1])); \\ Michel Marcus, Jun 16 2021

Formula

a(p) = 1 for p prime.

a(n) = Sum_{d|n} d^omega(n/d) * c(d), where c = A010051. - Wesley Ivan Hurt, Apr 13 2025