A344931 Sum of the distinct even-indexed prime divisors, p_{2k}, of n.

0, 0, 3, 0, 0, 3, 7, 0, 3, 0, 0, 3, 13, 7, 3, 0, 0, 3, 19, 0, 10, 0, 0, 3, 0, 13, 3, 7, 29, 3, 0, 0, 3, 0, 7, 3, 37, 19, 16, 0, 0, 10, 43, 0, 3, 0, 0, 3, 7, 0, 3, 13, 53, 3, 0, 7, 22, 29, 0, 3, 61, 0, 10, 0, 13, 3, 0, 0, 3, 7, 71, 3, 0, 37, 3, 19, 7, 16, 79, 0, 3, 0, 0, 10, 0, 43, 32

Offset: 1

Author

Wesley Ivan Hurt, Jun 02 2021

Keywords

• nonn

Comments

Inverse Möbius transform of n * c(n) * ((pi(n)+1) mod 2), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 23 2024

Examples

			a(12) = Sum_{p|12} p * ((pi(p)+1) mod 2) = 2*0 + 3*1 = 3.
		

Links

• Martin Ehrenstein, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000720 (pi), A008472 (sopf), A005074, A324966.

Cf. A344908 (sum of distinct odd-indexed prime divisors).

Programs

Mathematica

Table[Sum[k*Mod[PrimePi[k] + 1, 2] (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~, if (!(primepi(f[k,1]) % 2), f[k,1])); \\ Michel Marcus, Jun 12 2021

Formula

a(n) = Sum_{p|n} p * ((pi(p)+1) mod 2).

G.f.: Sum_{k>=1} prime(2*k) * x^prime(2*k) / (1 - x^prime(2*k)). - Ilya Gutkovskiy, Oct 24 2023

a(n) = Sum_{d|n} d * c(d) * ((pi(d)+1) mod 2), where c = A010051. - Wesley Ivan Hurt, Jun 23 2024