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A064604 Partial sums of A001159: Sum_{j=1..n} sigma_4(j).

%I A064604 #37 Oct 21 2023 15:51:22
%S A064604 1,18,100,373,999,2393,4795,9164,15807,26449,41091,63477,92039,132873,
%T A064604 184205,254110,337632,450563,580885,751783,948747,1197661,1477503,
%U A064604 1835761,2227012,2712566,3250650,3906396,4613678,5486322,6409844
%N A064604 Partial sums of A001159: Sum_{j=1..n} sigma_4(j).
%C A064604 In general, Sum_{k=1..n} sigma_j(k) = Sum_{k=1..n} (Bernoulli(j+1, floor(1 + n/k)) - Bernoulli(j+1, 0))/(j+1), where Bernoulli(n,x) are the Bernoulli polynomials, for any positive integer j. - _Daniel Suteu_, Nov 07 2018
%H A064604 Robert Israel, <a href="/A064604/b064604.txt">Table of n, a(n) for n = 1..10000</a>
%F A064604 a(n) = a(n-1) + A001159(n) = Sum_{j=1..n} sigma_4(j), where sigma_4(j) = A001159(j).
%F A064604 G.f.: (1/(1 - x))*Sum_{k>=1} k^4*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Jan 23 2017
%F A064604 a(n) ~ zeta(5) * n^5 / 5. - _Vaclav Kotesovec_, Sep 02 2018
%F A064604 a(n) = Sum_{k=1..n} Bernoulli(5, floor(1 + n/k))/5, where Bernoulli(n,x) are the Bernoulli polynomials. - _Daniel Suteu_, Nov 07 2018
%F A064604 a(n) = Sum_{k=1..n} k^4 * floor(n/k). - _Daniel Suteu_, Nov 08 2018
%p A064604 ListTools:-PartialSums(map(numtheory:-sigma[4],[$1..100])); # _Robert Israel_, Jun 29 2018
%t A064604 Accumulate[DivisorSigma[4, Range[50]]] (* _Vaclav Kotesovec_, Mar 30 2018 *)
%o A064604 (PARI) vector(50, n, sum(j=1, n, sigma(j,4))) \\ _G. C. Greubel_, Nov 07 2018
%o A064604 (Magma) [(&+[DivisorSigma(4,j): j in [1..n]]): n in [1..50]]; // _G. C. Greubel_, Nov 07 2018
%o A064604 (Python)
%o A064604 from math import isqrt
%o A064604 def A064604(n): return (-(s:=isqrt(n))**2*(s**2*(s*(6*s+15)+10)-1) + sum((q:=n//k)*(30*k**4+q**2*(q*(6*q+15)+10)-1) for k in range(1,s+1)))//30 # _Chai Wah Wu_, Oct 21 2023
%Y A064604 Cf. A001159, A064607.
%K A064604 nonn
%O A064604 1,2
%A A064604 _Labos Elemer_, Sep 24 2001