A333842 G.f.: Sum_{k>=1} k * x^(prime(k)^2) / (1 - x^(prime(k)^2)).

0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 3, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 4, 3, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 3, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 4, 2, 4, 0, 0, 0, 1

Offset: 1

Author

Ilya Gutkovskiy, Apr 07 2020

Keywords

• nonn

Comments

Sum of indices of non-unitary prime factors of n (prime factors for which the exponent exceeds 1).

Examples

			a(450) = a(2 * 3^2 * 5^2) = a(prime(1) * prime(2)^2 * prime(3)^2) = 2 + 3 = 5.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from indices in prime factorization.

Crossrefs

Cf. A000720, A003557, A005117 (positions of 0's), A056170, A056239, A063958, A066328, A071773.

Programs

Mathematica

nmax = 104; CoefficientList[Series[Sum[k x^(Prime[k]^2)/(1 - x^(Prime[k]^2)), {k, 1, nmax}], {x,0, nmax}], x] // Rest
f[p_, e_] := If[e == 1, 0, PrimePi[p]]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)

PARI

A333842(n) = { my(f=factor(n)); sum(k=1, #f~, if(1==f[k, 2],0,1)*primepi(f[k, 1])); }; \\ Antti Karttunen, Jun 12 2020

Formula

a(n) = A056239(A071773(n)) = A066328(A003557(n)). - Peter Munn and Antti Karttunen, Jun 13 2020

Additive with a(p^e) = primepi(p) = A000720(p) if e >= 2, and 0 otherwise. - Amiram Eldar, Jul 24 2024