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A338165
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Dirichlet g.f.: (zeta(s-3) / zeta(s))^2.
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1
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1, 14, 52, 161, 248, 728, 684, 1680, 2080, 3472, 2660, 8372, 4392, 9576, 12896, 16576, 9824, 29120, 13716, 39928, 35568, 37240, 24332, 87360, 46376, 61488, 74412, 110124, 48776, 180544, 59580, 157696, 138320, 137536, 169632, 334880, 101304, 192024, 228384, 416640
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OFFSET
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1,2
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COMMENTS
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Dirichlet convolution of Jordan function J_3 (A059376) with itself.
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LINKS
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FORMULA
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Multiplicative with a(p^e) = p^(3*e - 6) * (p^6 + e * (p^3 - 1)^2 - 1).
a(n) = Sum_{d|n} J_3(d) * J_3(n/d).
(1/tau(n)) * Sum_{d|n} a(d) * tau(n/d) = n^3.
Sum_{k=1..n} a(k) ~ 2025 * n^4 * ((log(n) + 2*gamma - 1/4)/Pi^8 - 180*zeta'(4) / Pi^12), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 14 2020
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MATHEMATICA
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Jordan3[n_] := Sum[d^3 MoebiusMu[n/d], {d, Divisors[n]}]; a[n_] := Sum[Jordan3[d] Jordan3[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 40}]
a[1] = 1; f[p_, e_] := p^(3 e - 6) (p^6 + e (p^3 - 1)^2 - 1); a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 40}]
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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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