cp's OEIS Frontend

A356249 a(n) = Sum_{k=1..n} (k * floor(n/k))^3.

%I A356249 #28 Oct 21 2023 18:00:33
%S A356249 1,16,62,219,405,1053,1523,2948,4407,7041,8703,15283,17949,24657,
%T A356249 32685,44806,50536,70687,78573,105411,125879,149879,163565,222425,
%U A356249 247476,286134,327634,396258,423084,532236,564818,664763,738095,821693,904937,1107618,1162268,1277588,1395760
%N A356249 a(n) = Sum_{k=1..n} (k * floor(n/k))^3.
%H A356249 Seiichi Manyama, <a href="/A356249/b356249.txt">Table of n, a(n) for n = 1..10000</a>
%F A356249 a(n) = Sum_{k=1..n} k^3 * Sum_{d|k} (1 - (1 - 1/d)^3).
%F A356249 G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k - 1)^3) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^4.
%F A356249 From _Vaclav Kotesovec_, Aug 02 2022: (Start)
%F A356249 a(n) = A064603(n) - 3*A356125(n) + 3*A319086(n).
%F A356249 a(n) ~ n^4 * (Pi^2/8 + Pi^4/360 - 3*zeta(3)/4). (End)
%t A356249 a[n_] := Sum[(k * Floor[n/k])^3, {k, 1, n}]; Array[a, 40] (* _Amiram Eldar_, Jul 31 2022 *)
%o A356249 (PARI) a(n) = sum(k=1, n, (k*(n\k))^3);
%o A356249 (PARI) a(n) = sum(k=1, n, k^3*sumdiv(k, d, 1-(1-1/d)^3));
%o A356249 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^4)/(1-x))
%o A356249 (Python)
%o A356249 from math import isqrt
%o A356249 def A356249(n): return -(s:=isqrt(n))**5*(s+1)**2 + sum((q:=n//k)**2*(k*(3*(k-1))+q*(k*(k*(4*k+6)-6)+q*(k*(3*(k-1))+1)+2)+1) for k in range(1,s+1))>>2 # _Chai Wah Wu_, Oct 21 2023
%Y A356249 Column k=3 of A356250.
%Y A356249 Cf. A000537, A024916, A318742, A350123.
%Y A356249 Cf. A064603, A356125, A319086.
%K A356249 nonn
%O A356249 1,2
%A A356249 _Seiichi Manyama_, Jul 31 2022