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A057523
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Number of ways n-th powerful (1) number can be expressed as b^2*c^3.
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2
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1
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OFFSET
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1,11
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COMMENTS
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Always positive. (Would be zero for any non-powerful number.)
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LINKS
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FORMULA
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Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3). (End)
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EXAMPLE
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a(40) = 2 since the 40th powerful number is 576 = 24^2*1^3 = 3^2*4^3.
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MATHEMATICA
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f[p_, e_] := Floor[(e + 2)/2] - Floor[(e + 2)/3]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 10000], # > 0 &] (* Amiram Eldar, Feb 23 2024 *)
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PROG
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(PARI) lista(kmax) = {my(e); for(k = 1, kmax, e = factor(k)[, 2]; if(k == 1 || vecmin(e) > 1, print1(vecprod(apply(x -> (x+2)\2 - (x+2)\3, e)), ", "))); } \\ Amiram Eldar, Feb 23 2024
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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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STATUS
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approved
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