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A360721
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a(n) is the number of infinitary divisors of n that are powerful (A001694).
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4
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1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 6, 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^e) = 2^A000120(e) - (e mod 2).
a(n) <= A037445(n) with equality if and only if n is a square.
Sum_{k=1..n} a(k) a(k) ~ c * n, where c = Product_{p prime} ((1-1/p) * Sum_{k>=1} ((2^A000120(k)- k mod 2)/p^k)) = 1.72717... .
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MATHEMATICA
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f[p_, e_] := 2^DigitCount[e, 2, 1] - Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
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PROG
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(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, 2^hammingweight(f[i, 2]) - f[i, 2]%2); }
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CROSSREFS
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Similar sequences: A005361 (number of powerful divisors), A323308 (number of unitary powerful divisors).
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KEYWORD
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nonn,mult,easy
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AUTHOR
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STATUS
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approved
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