A125030 - OEIS

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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

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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

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)

EXAMPLE

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

CROSSREFS

Cf. A077761, A001222, A125029, A125071.

Sequence in context: A325568 A326195 A073811 * A116479 A366989 A318322

Adjacent sequences: A125027 A125028 A125029 * A125031 A125032 A125033

KEYWORD

nonn,easy

AUTHOR

Leroy Quet, Nov 16 2006

EXTENSIONS

Extended by Ray Chandler, Nov 19 2006

STATUS

approved