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A034718 Dirichlet convolution of b_n=n with b_n with b_n. 7

%I

%S 1,6,9,24,15,54,21,80,54,90,33,216,39,126,135,240,51,324,57,360,189,

%T 198,69,720,150,234,270,504,87,810,93,672,297,306,315,1296,111,342,

%U 351,1200,123,1134,129,792,810,414,141,2160,294,900,459,936,159,1620,495

%N Dirichlet convolution of b_n=n with b_n with b_n.

%C Row sums of triangle A329323. - _Omar E. Pol_, Nov 21 2019

%H Amiram Eldar, <a href="/A034718/b034718.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{k*l*m = n} k*l*m, for positive integers k, l, m. This equals one sixth of the same sum over all integers. - _Ralf Stephan_, May 06 2005

%F Dirichlet g.f.: zeta^3(x-1).

%F Multiplicative with a(p^e) = p^e * binomial(e+2, 2). - _Mitch Harris_, Jun 27 2005

%F a(n) = n*A007425(n). Dirichlet convolution of A000027 by A038040. - _R. J. Mathar_, Mar 30 2011

%F Sum_{k=1..n} a(k) ~ (2*log(n)^2 + (12*gamma - 2)*log(n) + 12*gamma^2 - 6*gamma - 12*sg1 + 1) * n^2 / 8, where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - _Vaclav Kotesovec_, Sep 11 2019

%F G.f.: Sum_{k>=1} k*tau(k)*x^k / (1 - x^k)^2, where tau = A000005. - _Ilya Gutkovskiy_, Sep 22 2020

%t Table[n*Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, 100}] (* _Vaclav Kotesovec_, Aug 31 2018 *)

%t f[p_, e_] := (e+1)*(e+2)*p^e/2; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Sep 29 2020 *)

%Y Cf. A000027, A001620, A007425, A038040, A082633, A329323.

%K nonn,mult,easy

%O 1,2

%A _Erich Friedman_

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Last modified April 11 22:17 EDT 2021. Contains 342895 sequences. (Running on oeis4.)