cp's OEIS Frontend

A318743 a(n) = Sum_{k=1..n} floor(n/k)^4.

%I A318743 #29 Oct 27 2023 09:57:23
%S A318743 1,17,83,274,644,1396,2502,4388,6919,10743,15385,22407,30233,41209,
%T A318743 53853,70650,88636,113308,138654,172332,207984,252416,298002,358654,
%U A318743 417873,492065,569061,663427,756053,875541,989063,1130915,1272967,1441383,1607147,1817080
%N A318743 a(n) = Sum_{k=1..n} floor(n/k)^4.
%H A318743 Seiichi Manyama, <a href="/A318743/b318743.txt">Table of n, a(n) for n = 1..10000</a>
%F A318743 a(n) = -A006218(n) + 4*A024916(n) - 6*A064602(n) + 4*A064603(n).
%F A318743 a(n) ~ zeta(4) * n^4.
%F A318743 a(n) ~ Pi^4 * n^4 / 90.
%F A318743 G.f.: (1/(1 - x)) * Sum_{k>=1} (2*k - 1) * (2*k^2 - 2*k + 1) * x^k/(1 - x^k). - _Ilya Gutkovskiy_, Jul 16 2019
%t A318743 Table[Sum[Floor[n/k]^4, {k, 1, n}], {n, 1, 40}]
%t A318743 Accumulate[Table[-DivisorSigma[0, k] + 4*DivisorSigma[1, k] - 6*DivisorSigma[2, k] + 4*DivisorSigma[3, k], {k, 1, 40}]]
%o A318743 (PARI) a(n) = sum(k=1, n, (n\k)^4); \\ _Michel Marcus_, Sep 03 2018
%o A318743 (Magma) [&+[Floor(n/k)^4:k in [1..n]]:n in [1..36]]; // _Marius A. Burtea_, Jul 16 2019
%o A318743 (Python)
%o A318743 from math import isqrt
%o A318743 def A318743(n): return -(s:=isqrt(n))**5+sum((q:=n//k)*(k**4-(k-1)**4+q**3) for k in range(1,s+1)) # _Chai Wah Wu_, Oct 26 2023
%Y A318743 Cf. A000005, A000203, A024916, A064602, A064603, A064604.
%Y A318743 Cf. A006218, A222548, A318742, A318744.
%K A318743 nonn
%O A318743 1,2
%A A318743 _Vaclav Kotesovec_, Sep 02 2018