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A368106
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The number of infinitary divisors of the powerful part of n.
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2
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1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 4, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1
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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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Multiplicative with a(p) = 1 and a(p^e) = 2^A000120(e) for e >= 2.
a(n) >= 1, with equality if and only if n is squarefree (A005117).
a(n) <= A037445(n), with equality if and only if n is powerful (A001694).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.89684906463124350536..., where f(x) = (1-x) * (Product_{k>=0} (1 + 2*x^(2^k)) - x).
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MATHEMATICA
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f[p_, e_] := If[e == 1, 1, 2^DigitCount[e, 2, 1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
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PROG
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(PARI) a(n) = vecprod(apply(x -> if(x == 1, 1, 2^hammingweight(x)), factor(n)[, 2]));
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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