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%I #26 Sep 12 2023 05:56:12
%S 1,6,8,23,12,48,16,72,42,72,24,184,28,96,96,201,36,252,40,276,128,144,
%T 48,576,98,168,184,368,60,576,64,522,192,216,192,966,76,240,224,864,
%U 84,768,88,552,504,288,96,1608,178,588,288,644,108,1104,288,1152,320,360
%N Dirichlet convolution of sigma(n) with itself.
%H T. D. Noe, <a href="/A034761/b034761.txt">Table of n, a(n) for n=1..10000</a>
%F Dirichlet g.f.: zeta^2(x)zeta^2(x-1).
%F 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]
%F Equals A134577 * A000005. - _Gary W. Adamson_, Nov 02 2007
%F Also the Dirichlet convolution A000005 by A038040. - _R. J. Mathar_, Apr 01 2011
%F 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
%t 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 *)
%Y Cf. A000005, A000203 (sigma), A001620, A073002, A074962, A134577.
%K nonn,mult
%O 1,2
%A _Erich Friedman_
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