A125070 - OEIS

A125070

a(n) = number of nonzero exponents in the prime factorization of n which are not primes.

0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 0, 2, 0, 1, 1, 3, 1, 0, 2, 2, 2, 0, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 2, 0, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 1, 0, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 0, 1, 3, 1, 1, 3

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OFFSET

1,6

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) = A005171(e).

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B - C), where B is Mertens's constant (A077761) and C = Sum_{p prime} (P(p) - P(p+1)) = 0.39847584805803104040..., where P(s) is the prime zeta function. (End)

EXAMPLE

720 has the prime-factorization of 2^4 *3^2 *5^1. Two of these exponents, 4 and 1, are not primes. So a(720) = 2.

MATHEMATICA

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

PROG

(PARI) A125070(n) = vecsum(apply(e -> if(isprime(e), 0, 1), factorint(n)[, 2])); \\ Antti Karttunen, Jul 07 2017

CROSSREFS

Cf. A001221, A005171, A077761, A125071.

Sequence in context: A286852 A341595 A369428 * A125071 A335699 A177207

Adjacent sequences: A125067 A125068 A125069 * A125071 A125072 A125073

KEYWORD

nonn,easy

AUTHOR

Leroy Quet, Nov 18 2006

EXTENSIONS

Extended by Ray Chandler, Nov 19 2006

STATUS

approved