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A366077
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The number of prime factors of the cubefree part of n (A360539), counted with multiplicity.
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5
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0, 1, 1, 2, 1, 2, 1, 0, 2, 2, 1, 3, 1, 2, 2, 0, 1, 3, 1, 3, 2, 2, 1, 1, 2, 2, 0, 3, 1, 3, 1, 0, 2, 2, 2, 4, 1, 2, 2, 1, 1, 3, 1, 3, 3, 2, 1, 1, 2, 3, 2, 3, 1, 1, 2, 1, 2, 2, 1, 4, 1, 2, 3, 0, 2, 3, 1, 3, 2, 3, 1, 2, 1, 2, 3, 3, 2, 3, 1, 1, 0, 2, 1, 4, 2, 2, 2
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OFFSET
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1,4
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COMMENTS
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The sum of exponents smaller than 3 in the prime factorization of n.
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LINKS
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FORMULA
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Additive with a(p^e) = e if e <= 2, and a(p^e) = 0 for e >= 3.
a(n) >= 0, with equality if and only if n is cubefull (A036966).
a(n) <= A001222(n), with equality if and only if n is cubefree (A004709).
a(n) >= A366078(n), with equality if and only if n is squarefree (A005117).
Sum_{k=1..m} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} (p-2)/p^3 = A085548 - 2 * A085541 = 0.10272214144217842566... .
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MATHEMATICA
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f[p_, e_] := If[e < 3, e, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
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PROG
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(PARI) a(n) = vecsum(apply(x -> if(x < 3, x, 0), factor(n)[, 2]));
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CROSSREFS
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Cf. A001222, A004709, A005117, A036966, A056169, A077761, A085541, A085548, A360539, A366076, A366078.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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