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A333843
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Expansion of g.f.: Sum_{k>=1} k * x^(k^3) / (1 - x^(k^3)).
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6
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1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 4, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4
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OFFSET
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1,8
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COMMENTS
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Sum of cube roots of cube divisors of n.
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LINKS
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FORMULA
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Dirichlet g.f.: zeta(s) * zeta(3*s-1).
If n = Product (p_j^k_j) then a(n) = Product ((p_j^(floor(k_j/3) + 1) - 1)/(p_j - 1)).
Sum_{k=1..n} a(k) ~ Pi^2*n/6 + zeta(2/3)*n^(2/3)/2. - Vaclav Kotesovec, Dec 01 2020
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MATHEMATICA
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nmax = 108; CoefficientList[Series[Sum[k x^(k^3)/(1 - x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, #^(1/3) &, IntegerQ[#^(1/3)] &], {n, 108}]
f[p_, e_] := (p^(Floor[e/3] + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 01 2020 *)
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PROG
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(PARI) a(n) = {my(f = factor(n)); prod(i=1, #f~, (f[i, 1]^(f[i, 2]\3 + 1) - 1)/(f[i, 1] - 1)); } \\ Amiram Eldar, Sep 05 2023
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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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