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A370256
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The number of ways in which n can be expressed as b^2 * c^3, with b and c >= 1.
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2
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1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0
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OFFSET
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1,64
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COMMENTS
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The least number k such that a(k) = n is A005179(n)^6.
The indices of records are the sixth powers of the highly composite numbers, A002182(n)^6.
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LINKS
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FORMULA
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Multiplicative with a(p^e) = A103221(e).
a(n) > 0 if and only if n is a powerful number (A001694).
Sum_{k=1..n} a(k) ~ zeta(3/2) * sqrt(n) + zeta(2/3) * n^(1/3).
Dirichlet generating function: zeta(2*s)*zeta(3*s). - Vaclav Kotesovec, Feb 23 2024
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EXAMPLE
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1 = 1^2 * 1^3, so a(1) = 1.
64 = 1^2 * 4^3 = 8^2 * 1^3, so a(64) = 2.
4096 = 64^2 * 1^3 = 8^2 * 4^3 = 1^2 * 16^3, so a(4096)= 3.
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MATHEMATICA
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f[p_, e_] := Floor[(e + 2)/2] - Floor[(e + 2)/3]; 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 -> (x+2)\2 - (x+2)\3, factor(n)[, 2]));
(PARI) for(n=1, 100, print1(direuler(p=2, n, 1/((1 - X^2)*(1 - X^3)))[n], ", ")) \\ Vaclav Kotesovec, Feb 23 2024
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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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