A123391 a(n) = sum of exponents that are primes in the prime factorization of n.

0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 3, 2, 0, 0, 0, 5, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 0, 2, 2, 0, 2, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 5, 0, 2, 2, 4, 0, 0, 0, 3, 0

Offset: 1

Author

Leroy Quet, Nov 10 2006

Keywords

• nonn
• easy

Examples

			36 = 2^2*3^2. Both exponents in this prime factorization are primes. So a(36) = 2+2 = 4.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A010051, A101436, A125030, A125071, A125073.

Programs

Mathematica

f[n_] := Plus @@ Select[Last /@ FactorInteger[n], PrimeQ];Table[f[n], {n, 120}] (* Ray Chandler, Nov 11 2006*)

PARI

{m=105; for(n=1, m, v=factor(n)[,2]; s=0; for(j=1, #v, if(isprime(v[j]), s=s+v[j])); print1(s,","))} \\ Klaus Brockhaus, Nov 14 2006

PARI

A123391(n) = vecsum(apply(e -> isprime(e)*e, factorint(n)[, 2])); \\ Antti Karttunen, Jul 19 2017

Formula

Additive with a(p^e) = A010051(e)*e. - Antti Karttunen, Jul 19 2017

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} p*(P(p)-P(p+1)) = 0.97487020987790163735..., where P(s) is the prime zeta function. - Amiram Eldar, Sep 29 2023

Extensions

Extended by Ray Chandler and Klaus Brockhaus, Nov 11 2006