A343394 Sum of indices of n's distinct prime factors below n.

0, 0, 0, 1, 0, 3, 0, 1, 2, 4, 0, 3, 0, 5, 5, 1, 0, 3, 0, 4, 6, 6, 0, 3, 3, 7, 2, 5, 0, 6, 0, 1, 7, 8, 7, 3, 0, 9, 8, 4, 0, 7, 0, 6, 5, 10, 0, 3, 4, 4, 9, 7, 0, 3, 8, 5, 10, 11, 0, 6, 0, 12, 6, 1, 9, 8, 0, 8, 11, 8, 0, 3, 0, 13, 5, 9, 9, 9, 0, 4, 2, 14, 0, 7, 10, 15, 12, 6, 0, 6

Offset: 1

Author

Ilya Gutkovskiy, Apr 13 2021

Keywords

• nonn

Examples

			a(7) = a(prime(4)) = 0.
a(21) = a(3 * 7) = a(prime(2) * prime(4)) = 2 + 4 = 6.
		

Links

• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A066328, A087624, A105221.

Programs

Mathematica

nmax = 90; CoefficientList[Series[Sum[k x^(2 Prime[k])/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := If[PrimeQ[n], 0, Plus @@ (PrimePi[#[[1]]] & /@ FactorInteger[n])]; Table[a[n], {n, 90}]

Formula

G.f.: Sum_{k>=1} k * x^(2*prime(k)) / (1 - x^prime(k)).

a(n) = 0 if n is prime, A066328(n) otherwise.