cp's OEIS Frontend

A318744 a(n) = Sum_{k=1..n} floor(n/k)^5.

%I A318744 #25 Oct 27 2023 09:57:19
%S A318744 1,33,245,1058,3160,8054,17086,33860,60353,103437,164489,257945,
%T A318744 380407,556001,779865,1085840,1457122,1958008,2544540,3312306,4205650,
%U A318744 5336264,6618976,8254674,10059777,12298021,14792045,17829881,21130663,25189011,29518163,34749419
%N A318744 a(n) = Sum_{k=1..n} floor(n/k)^5.
%H A318744 Seiichi Manyama, <a href="/A318744/b318744.txt">Table of n, a(n) for n = 1..10000</a>
%F A318744 a(n) = A006218(n) - 5*A024916(n) + 10*A064602(n) - 10*A064603(n) + 5*A064604(n).
%F A318744 a(n) ~ zeta(5) * n^5.
%t A318744 Table[Sum[Floor[n/k]^5, {k, 1, n}], {n, 1, 40}]
%t A318744 Accumulate[Table[DivisorSigma[0, k] - 5*DivisorSigma[1, k] + 10*DivisorSigma[2, k] - 10*DivisorSigma[3, k] + 5*DivisorSigma[4, k], {k, 1, 40}]]
%o A318744 (PARI) a(n) = sum(k=1, n, (n\k)^5); \\ _Michel Marcus_, Sep 03 2018
%o A318744 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (k^5-(k-1)^5)*x^k/(1-x^k))/(1-x)) \\ _Seiichi Manyama_, May 27 2021
%o A318744 (Python)
%o A318744 from math import isqrt
%o A318744 def A318744(n): return -(s:=isqrt(n))**6+sum((q:=n//k)*(k**5-(k-1)**5+q**4) for k in range(1,s+1)) # _Chai Wah Wu_, Oct 26 2023
%Y A318744 Cf. A000005, A000203, A024916, A064602, A064603, A064604.
%Y A318744 Cf. A006218, A222548, A318742, A318743.
%K A318744 nonn
%O A318744 1,2
%A A318744 _Vaclav Kotesovec_, Sep 02 2018