A344908 -id:A344908 - cp's OEIS frontend

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

Wesley Ivan Hurt, Jun 02 2021

nonn

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

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

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

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

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

Table[Sum[k*Mod[PrimePi[k] + 1, 2] (PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]

a(n) = my(f=factor(n)); sum(k=1, #f~, if (!(primepi(f[k,1]) % 2), f[k,1])); \\ Michel Marcus, Jun 12 2021

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

A366725 Sum of odd indices of distinct prime factors of n.

Original entry on oeis.org

0, 1, 0, 1, 3, 1, 0, 1, 0, 4, 5, 1, 0, 1, 3, 1, 7, 1, 0, 4, 0, 6, 9, 1, 3, 1, 0, 1, 0, 4, 11, 1, 5, 8, 3, 1, 0, 1, 0, 4, 13, 1, 0, 6, 3, 10, 15, 1, 0, 4, 7, 1, 0, 1, 8, 1, 0, 1, 17, 4, 0, 12, 0, 1, 3, 6, 19, 8, 9, 4, 0, 1, 21, 1, 3, 1, 5, 1, 0, 4, 0, 14, 23, 1, 10, 1, 0, 6, 0, 4, 0, 10, 11, 16, 3, 1, 25, 1, 5, 4

Offset: 1

Ilya Gutkovskiy, Oct 24 2023

nonn

			a(60) = 4 because 60 = 2^2 * 3 * 5 = prime(1)^2 * prime(2) * prime(3) and 1 + 3 = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000720 (pi), A066207 (positions of 0's), A066328, A324966, A332422, A344908, A366528, A366784.

nmax = 100; CoefficientList[Series[Sum[(2 k - 1) x^Prime[2 k - 1]/(1 - x^Prime[2 k - 1]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := Module[{i = PrimePi[p]}, If[OddQ[i], i, 0]]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 03 2025 *)

f(n) = if(n % 2, n, 0);
a(n) = vecsum(apply(x -> f(primepi(x)), factor(n)[, 1])); \\ Amiram Eldar, Jul 03 2025

G.f.: Sum_{k>=1} (2*k-1) * x^prime(2*k-1) / (1 - x^prime(2*k-1)).
From Amiram Eldar, Jul 03 2025: (Start)
Additive with a(p^e) = pi(p) if pi(p) is odd, and 0 otherwise.
a(n) = A066328(n) - 2*A366784(n). (End)

cp's OEIS Frontend

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

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