A366784 Sum of even indices of distinct prime factors of n divided by 2.
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
Keywords
Examples
a(315) = 3 because 315 = 3^2 * 5 * 7 = prime(2)^2 * prime(3) * prime(4) and (2 + 4) / 2 = 3.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Crossrefs
Programs
-
Mathematica
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 *) -
PARI
f(n) = if(n % 2, 0, n/2); a(n) = vecsum(apply(x -> f(primepi(x)), factor(n)[, 1])); \\ Amiram Eldar, Jul 03 2025
Formula
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.