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%I #39 Sep 08 2022 08:45:03
%S 1,6,7,21,9,42,11,58,30,54,15,147,17,66,63,141,21,180,23,189,77,90,27,
%T 406,54,102,106,231,33,378,35,318,105,126,99,630,41,138,119,522,45,
%U 462,47,315,270,162,51,987,86,324,147,357,57,636,135,638,161,198,63,1323
%N Dirichlet convolution of n and tau^2(n).
%C Dirichlet convolution of A000027 and A035116.
%C Inverse Mobius transform of A060724. - _R. J. Mathar_, Oct 15 2011
%H Vincenzo Librandi, <a href="/A062369/b062369.txt">Table of n, a(n) for n = 1..5000</a>
%F a(n) = Sum_{i|n, j|n} sigma(i)*sigma(j)/sigma(lcm(i,j)), where sigma(n) = sum of divisors of n.
%F a(n) = Sum_{i|d, j|d} sigma(gcd(i, j));
%F a(n) = Sum_{d|n} d*tau(n/d)^2, where tau(n) = number of divisors of n.
%F Multiplicative with a(p^e) = (1-p^(3+e)-p^(2+e)+e^2+4*p^2+p^2*e^2+2*e-3*p+4*p^2*e-6*e*p-2*e^2*p)/(1-p)^3.
%F Dirichlet g.f.: (zeta(s))^4*zeta(s-1)/zeta(2*s). - _R. J. Mathar_, Feb 09 2011
%F G.f.: Sum_{k>=1} tau(k)^2*x^k/(1 - x^k)^2. - _Ilya Gutkovskiy_, Nov 02 2018
%F Sum_{k=1..n} a(k) ~ 5 * Pi^4 * n^2 / 144. - _Vaclav Kotesovec_, Jan 28 2019
%F a(n) = Sum_{d|n} tau(d^2)*sigma(n/d), where tau(n) = number of divisors of n, and sigma(n) = sum of divisors of n. - _Ridouane Oudra_, Aug 25 2019
%t a[n_] := Sum[ DivisorSigma[1, i]*DivisorSigma[1, j] / DivisorSigma[1, LCM[i, j]], {i, Divisors[n]}, {j, Divisors[n]}]; Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Mar 26 2013 *)
%o (PARI) a(n) = sumdiv(n, d, d*numdiv(n/d)^2); \\ _Michel Marcus_, Nov 03 2018
%o (Magma) [&+[d*#Divisors(Floor(n/d))^2:d in Divisors(n)]:n in [1..60]]; // _Marius A. Burtea_, Aug 25 2019
%Y Cf. A000203, A060648.
%Y Cf. A000005, A060724, A064950, A062369, A062368, A062380.
%K nonn,mult
%O 1,2
%A _Vladeta Jovovic_, Jul 07 2001
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