A344931 - OEIS

%I #35 Jun 23 2024 01:45:58

%S 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,

%T 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,

%U 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

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

%C 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

%H Martin Ehrenstein, <a href="/A344931/b344931.txt">Table of n, a(n) for n = 1..20000</a>

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

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

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

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

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

%o (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

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

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

%K nonn

%O 1,3

%A _Wesley Ivan Hurt_, Jun 02 2021