A344908 Sum of the distinct odd-indexed prime divisors, p_{2k-1}, of n.

0, 2, 0, 2, 5, 2, 0, 2, 0, 7, 11, 2, 0, 2, 5, 2, 17, 2, 0, 7, 0, 13, 23, 2, 5, 2, 0, 2, 0, 7, 31, 2, 11, 19, 5, 2, 0, 2, 0, 7, 41, 2, 0, 13, 5, 25, 47, 2, 0, 7, 17, 2, 0, 2, 16, 2, 0, 2, 59, 7, 0, 33, 0, 2, 5, 13, 67, 19, 23, 7, 0, 2, 73, 2, 5, 2, 11, 2, 0, 7, 0, 43, 83, 2, 22

Offset: 1

Author

Wesley Ivan Hurt, Jun 02 2021

Keywords

• nonn

Comments

a(m) = 0 for m in A066207. - Michel Marcus, Jun 12 2021

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

Examples

			a(6) = Sum_{p|6} p * (pi(p) mod 2) = 2*(pi(2) mod 2) + 3*(pi(3) mod 2) = 2*1 + 3*0 = 2.
		

Links

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

Crossrefs

Cf. A000720 (pi), A008472 (sopf), A005074, A010051, A066207, A324966.

Cf. A344931 (sum of distinct even-indexed prime divisors).

Programs

Mathematica

Table[Sum[k*Mod[PrimePi[k], 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) mod 2).

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

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