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A333844
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G.f.: Sum_{k>=1} k * x^(k^4) / (1 - x^(k^4)).
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3
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3
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OFFSET
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1,16
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COMMENTS
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Sum of 4th roots of 4th powers dividing n.
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LINKS
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FORMULA
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Dirichlet g.f.: zeta(s) * zeta(4*s-1).
If n = Product (p_j^k_j) then a(n) = Product ((p_j^(floor(k_j/4) + 1) - 1)/(p_j - 1)).
Sum_{k=1..n} a(k) ~ zeta(3)*n + zeta(1/2)*sqrt(n)/2. - Vaclav Kotesovec, Dec 01 2020
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MATHEMATICA
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nmax = 112; CoefficientList[Series[Sum[k x^(k^4)/(1 - x^(k^4)), {k, 1, Floor[nmax^(1/4)] + 1}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, #^(1/4) &, IntegerQ[#^(1/4)] &], {n, 112}]
f[p_, e_] := (p^(Floor[e/4] + 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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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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