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A386013 a(n) = n^4*tau(n).

%I A386013 #17 Aug 03 2025 06:43:56
%S A386013 1,32,162,768,1250,5184,4802,16384,19683,40000,29282,124416,57122,
%T A386013 153664,202500,327680,167042,629856,260642,960000,777924,937024,
%U A386013 559682,2654208,1171875,1827904,2125764,3687936,1414562,6480000,1847042,6291456,4743684,5345344,6002500,15116544,3748322,8340544,9253764,20480000
%N A386013 a(n) = n^4*tau(n).
%C A386013 Dirichlet convolution of the 4th powers A000583 with themselves.
%H A386013 Amiram Eldar, <a href="/A386013/b386013.txt">Table of n, a(n) for n = 1..10000</a>
%F A386013 a(n) = A000005(n) * A000583(n).
%F A386013 a(n) = n^2*A034714(n) = n^3*A038040(n) = n*A386012(n).
%F A386013 Dirichlet g.f.: zeta^2(s-4).
%F A386013 From _Amiram Eldar_, Jul 15 2025 (Start)
%F A386013 Multiplicative with a(p^e) = p^(4*e) * (e+1).
%F A386013 Sum_{k=1..n} a(k) ~ (n^5/5) * (log(n) + 2*gamma - 1/5), where gamma is Euler's constant (A001620). (End)
%F A386013 G.f.: Sum_{k>=1} k^4*x^k*(1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(1 - x^k)^5. - _Vaclav Kotesovec_, Aug 03 2025
%p A386013 seq( n^4*numtheory[tau](n),n=1..100) ;
%t A386013 a[n_]:=n^4*DivisorSigma[0,n]; Array[a,40] (* _Stefano Spezia_, Jul 14 2025 *)
%t A386013 nmax = 40; Rest[CoefficientList[Series[Sum[k^4*x^k*(1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(1 - x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Aug 03 2025 *)
%o A386013 (PARI) a(n) = n^4 * numdiv(n); \\ _Amiram Eldar_, Jul 15 2025
%Y A386013 Cf. A000005, A000583, A001620, A034714, A038040, A372931 (Mobius transform).
%K A386013 nonn,mult,easy
%O A386013 1,2
%A A386013 _R. J. Mathar_, Jul 14 2025