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a(n) = Sum_{d|n} d*sigma(n/d)*sigma(d).
2

%I #22 Aug 27 2023 02:52:23

%S 1,9,16,53,36,144,64,261,178,324,144,848,196,576,576,1173,324,1602,

%T 400,1908,1024,1296,576,4176,986,1764,1744,3392,900,5184,1024,5013,

%U 2304,2916,2304,9434,1444,3600,3136,9396,1764,9216,1936,7632,6408,5184,2304

%N a(n) = Sum_{d|n} d*sigma(n/d)*sigma(d).

%C Logarithmic derivative of A174467.

%C Dirichlet convolution of A064987 and A000203. - _R. J. Mathar_, Feb 06 2011

%H Antti Karttunen, <a href="/A174468/b174468.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = sigma(n)^2 iff n is squarefree.

%F Dirichlet g.f. zeta(s) * (zeta(s-1))^2 * zeta(s-2). - _R. J. Mathar_, Feb 06 2011

%F Sum_{k=1..n} a(k) ~ Pi^4 * zeta(3) * n^3 / 108. - _Vaclav Kotesovec_, Feb 02 2019

%F Multiplicative with a(p^e) = (p^(2*e+4) - (e+2)*(p^2-1)*p^(e+1) - 1)/((p-1)^3*(p+1)). - _Amiram Eldar_, Aug 27 2023

%t Table[Sum[d*DivisorSigma[1,d]*DivisorSigma[1,n/d], {d, Divisors[n]}], {n, 1, 50}] (* _Vaclav Kotesovec_, Feb 02 2019 *)

%t f[p_, e_] := (p^(2*e + 4) - (e + 2)*(p^2 - 1)*p^(e + 1) - 1)/((p - 1)^3*(p + 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Aug 27 2023 *)

%o (PARI) {a(n)=sumdiv(n,d,d*sigma(n/d)*sigma(d))}

%o (PARI) a(n)=sumdiv(n, x, x * sumdiv(x, y, sumdiv(y, z, z ) ) ); /* _Joerg Arndt_, Oct 07 2012 */

%Y Cf. A174467, A000203 (sigma), A064987.

%K nonn,easy,mult

%O 1,2

%A _Paul D. Hanna_, Apr 04 2010