A356126 - OEIS

%I #20 Oct 24 2023 15:42:28

%S 1,19,103,395,1025,2537,4945,9625,16438,27778,42430,66958,95532,

%T 138876,191796,266692,350230,472864,603204,787164,989436,1253172,

%U 1533036,1926156,2319931,2834263,3386143,4089279,4796589,5749149,6672701,7871069,9101837,10605521

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

%H Seiichi Manyama, <a href="/A356126/b356126.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{k=1..n} k^4 * binomial(floor(n/k)+1,2).

%F G.f.: (1/(1-x)) * Sum_{k>=1} k^4 * x^k/(1 - x^k)^2.

%t a[n_] := Sum[k * DivisorSigma[3, k], {k, 1, n}]; Array[a, 34] (* _Amiram Eldar_, Jul 28 2022 *)

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

%o (PARI) a(n) = sum(k=1, n, k^4*binomial(n\k+1, 2));

%o (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k^4*x^k/(1-x^k)^2)/(1-x))

%o (Python)

%o from math import isqrt

%o def A356126(n): return ((-(s:=isqrt(n))**2*(s+1)**2*((s<<1)+1)*(s*(3*(s+1))-1)>>1)+sum(k*(q:=n//k)*(q+1)*(15*k**3+((q<<1)+1)*(q*(3*(q+1))-1)) for k in range(1,s+1)))//30 # _Chai Wah Wu_, Oct 24 2023

%Y Partial sums of A281372.

%Y Column k=4 of A356124.

%Y Cf. A356043.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Jul 27 2022