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A034761
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Dirichlet convolution of sigma(n) with itself.
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10
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1, 6, 8, 23, 12, 48, 16, 72, 42, 72, 24, 184, 28, 96, 96, 201, 36, 252, 40, 276, 128, 144, 48, 576, 98, 168, 184, 368, 60, 576, 64, 522, 192, 216, 192, 966, 76, 240, 224, 864, 84, 768, 88, 552, 504, 288, 96, 1608, 178, 588, 288, 644, 108, 1104, 288, 1152, 320, 360
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OFFSET
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1,2
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LINKS
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FORMULA
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Dirichlet g.f.: zeta^2(x)zeta^2(x-1).
Multiplicative with a(2^e) = (e-1) 2^(e+2) + e + 5, a(p^e) = ((1+e)p^(e+3) - (3+e)(p^(e+2)-p+1) + 2)/(p-1)^3, p > 2. - Mitch Harris, Jun 27 2005 [corrected by Amiram Eldar, Oct 16 2022 and Sep 12 2023]
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 * (2*Pi^2 * log(n) + (4*gamma - 1)*Pi^2 + 24*zeta'(2)) / 144, where gamma is the Euler-Mascheroni constant A001620 and Zeta'(2) = A073002. Equivalently, Sum_{k=1..n} a(k) ~ Pi^4 * n^2 * (2*log(n) - 1 + 8*gamma - 48*log(A) + 4*log(2*Pi)) / 144, where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jan 28 2019
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MATHEMATICA
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f[p_, e_] := ((e + 1)*p^(e + 3) - (e + 3)*(p^(e + 2) - p + 1) + 2)/(p - 1)^3; f[2, e_] := (e - 1)*2^(e + 2) + e + 5; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Oct 16 2022 *)
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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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