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

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n.

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

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

Sequence in context: A082507 A132349 A216226 * A375339 A372603 A212172

Adjacent sequences: A123388 A123389 A123390 * A123392 A123393 A123394

Keyword

nonn,easy

Author

Leroy Quet, Nov 10 2006

Extensions

Extended by Ray Chandler and Klaus Brockhaus, Nov 11 2006

Status

approved