a(720) = 3, since the prime factorization of 720 is 2^4 * 3^2 * 5^1 and two of the exponents in this factorization are noncomposites (the exponents 2 and 1, whose sum is 3).

MATHEMATICA

f[n_] := Plus @@ Select[Last /@ FactorInteger[n], # == 1 || PrimeQ[ # ] &]; Table[f[n], {n, 110}] (* Ray Chandler, Nov 19 2006 *)

PROG

(PARI) A125030(n) = vecsum(apply(e -> if((1==e)||isprime(e), e, 0), factorint(n)[, 2])); \\ Antti Karttunen, Jul 07 2017