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A125030
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a(n) = sum of exponents in the prime factorization of n that are noncomposite.
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4
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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
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1,4
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LINKS
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FORMULA
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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)
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EXAMPLE
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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).
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MATHEMATICA
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f[n_] := Plus @@ Select[Last /@ FactorInteger[n], # == 1 || PrimeQ[ # ] &]; Table[f[n], {n, 110}] (* Ray Chandler, Nov 19 2006 *)
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PROG
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(PARI) A125030(n) = vecsum(apply(e -> if((1==e)||isprime(e), e, 0), factorint(n)[, 2])); \\ Antti Karttunen, Jul 07 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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