A344931 -id:A344931 - cp's OEIS frontend

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

Wesley Ivan Hurt, Jun 02 2021

nonn

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

			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.

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

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

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

Table[Sum[k*Mod[PrimePi[k], 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) 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

A366784 Sum of even indices of distinct prime factors of n divided by 2.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 0, 1, 0, 0, 1, 3, 2, 1, 0, 0, 1, 4, 0, 3, 0, 0, 1, 0, 3, 1, 2, 5, 1, 0, 0, 1, 0, 2, 1, 6, 4, 4, 0, 0, 3, 7, 0, 1, 0, 0, 1, 2, 0, 1, 3, 8, 1, 0, 2, 5, 5, 0, 1, 9, 0, 3, 0, 3, 1, 0, 0, 1, 2, 10, 1, 0, 6, 1, 4, 2, 4, 11, 0, 1, 0, 0, 3, 0, 7, 6, 0, 12, 1, 5, 0, 1, 0, 4, 1, 0, 2, 1

Offset: 1

Ilya Gutkovskiy, Oct 24 2023

nonn

			a(315) = 3 because 315 = 3^2 * 5 * 7 = prime(2)^2 * prime(3) * prime(4) and (2 + 4) / 2 = 3.

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

Cf. A000720 (pi), A066208 (positions of 0's), A066328, A324967, A332422, A344931, A366533, A366725.

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

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

G.f.: Sum_{k>=1} k * x^prime(2*k) / (1 - x^prime(2*k)).
From Amiram Eldar, Jul 03 2025: (Start)
Additive with a(p^e) = pi(p)/2 if pi(p) is even, and 0 otherwise.
a(n) = (A066328(n) - A366725(n))/2. (End)

cp's OEIS Frontend

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

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