A350108 - OEIS

%I #30 Oct 31 2023 18:14:25

%S 1,10,32,87,153,309,443,722,1005,1443,1785,2605,3087,3951,4875,6154,

%T 6988,8809,9855,12057,13853,16001,17543,21347,23478,26484,29440,33696,

%U 36162,41994,44816,50351,54755,59909,64577,73524,77558,84002,90142,100072,105034

%N a(n) = Sum_{k=1..n} k * floor(n/k)^3.

%H Seiichi Manyama, <a href="/A350108/b350108.txt">Table of n, a(n) for n = 1..10000</a>

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

%F G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k - 1)^3) * x^k/(1 - x^k)^2.

%F From _Vaclav Kotesovec_, Aug 03 2022: (Start)

%F a(n) = A024916(n) + 3*A143128(n) - 3*A143127(n).

%F a(n) ~ Pi^2*n^3/6 - 3*n^2*log(n)/2. (End)

%t a[n_] := Sum[k * Floor[n/k]^3, {k, 1, n}]; Array[a, 40] (* _Amiram Eldar_, Dec 14 2021 *)

%t Accumulate[Table[(1 + 3*k)*DivisorSigma[1, k] - 3*k*DivisorSigma[0, k], {k, 1, 50}]] (* _Vaclav Kotesovec_, Dec 16 2021 *)

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

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

%o (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k/(1-x^k)^2)/(1-x))

%o (Python)

%o from math import isqrt

%o def A350108(n): return -(s:=isqrt(n))**4*(s+1)+sum((q:=n//k)*(k**2*(3*(q+1))+k*(q*((q<<1)-3)-3)+q+1) for k in range(1,s+1))>>1 # _Chai Wah Wu_, Oct 31 2023

%Y Column 3 of A350106.

%Y Cf. A024916, A143128, A143127, A318742.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Dec 14 2021