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A123391
a(n) = sum of exponents that are primes in the prime factorization of n.
2
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
1,4
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
EXAMPLE
36 = 2^2*3^2. Both exponents in this prime factorization are primes. So a(36) = 2+2 = 4.
MATHEMATICA
f[n_] := Plus @@ Select[Last /@ FactorInteger[n], PrimeQ]; Table[f[n], {n, 120}] (* Ray Chandler, Nov 11 2006*)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Leroy Quet, Nov 10 2006
EXTENSIONS
Extended by Ray Chandler and Klaus Brockhaus, Nov 11 2006
STATUS
approved