A356043 - OEIS

%I #14 Aug 07 2022 04:15:02

%S 1,11,40,123,250,540,885,1553,2339,3609,4942,7349,9548,12998,16681,

%T 22030,26945,34805,41666,52207,62212,75542,87711,107083,122961,144951,

%U 166177,194812,219203,256033,285826,328624,367281,416431,460246,525484,576139,644749,708520

%N a(n) = Sum_{k=1..n} sigma_3(k) * floor(n/k).

%F a(n) = Sum_{k=1..n} Sum_{d|k} d^3 * tau(k/d).

%F G.f.: (1/(1-x)) * Sum_{k>=1} sigma_3(k) * x^k/(1 - x^k).

%F a(n) ~ Pi^8 * n^4 / 32400. - _Vaclav Kotesovec_, Aug 07 2022

%t Table[Sum[DivisorSigma[3, k]*Floor[n/k], {k, 1, n}], {n, 1, 50}] (* _Vaclav Kotesovec_, Aug 07 2022 *)

%o (PARI) a(n) = sum(k=1, n, sigma(k, 3)*(n\k));

%o (PARI) a(n) = sum(k=1, n, sumdiv(k, d, d^3*numdiv(k/d)));

%o (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k, 3)*x^k/(1-x^k))/(1-x))

%Y Partial sums of A321140.

%Y Column k=3 of A356045.

%Y Cf. A000005 (tau).

%K nonn

%O 1,2

%A _Seiichi Manyama_, Jul 24 2022