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A125070
a(n) = number of nonzero exponents in the prime factorization of n which are not primes.
9
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
OFFSET
1,6
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
KEYWORD
nonn,easy
AUTHOR
Leroy Quet, Nov 18 2006
EXTENSIONS
Extended by Ray Chandler, Nov 19 2006
STATUS
approved