A125030 a(n) = sum of exponents in the prime factorization of n that are noncomposite.

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 0, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 1, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 4, 1, 2, 3, 0, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 1, 0, 2, 1, 4, 2, 2, 2, 4, 1, 4, 2, 3, 2, 2, 2, 6, 1, 3, 3, 4, 1, 3, 1, 4, 3

Offset: 1

Author

Leroy Quet, Nov 16 2006

Keywords

• nonn
• easy

Examples

			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).
		

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. A077761, A001222, A125029, A125071.

Programs

Mathematica

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

PARI

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

Formula

From Amiram Eldar, Sep 30 2023: (Start)

Additive with a(p^e) = e if e is composite, and 0 otherwise.

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = - P(2) + Sum_{p prime} p * (P(p) - P(p+1)) = 0.52262278983683613884..., where P(s) is the prime zeta function. (End)

Extensions

Extended by Ray Chandler, Nov 19 2006