cp's OEIS Frontend

A248076 Partial sums of the sum of the 5th powers of the divisors of n: Sum_{i=1..n} sigma_5(i).

%I A248076 #44 Oct 22 2023 00:35:16
%S A248076 1,34,278,1335,4461,12513,29321,63146,122439,225597,386649,644557,
%T A248076 1015851,1570515,2333259,3415660,4835518,6792187,9268287,12572469,
%U A248076 16673621,21988337,28424681,36677981,46446732,58699434,73107634,90873690,111384840,136555392
%N A248076 Partial sums of the sum of the 5th powers of the divisors of n: Sum_{i=1..n} sigma_5(i).
%H A248076 Seiichi Manyama, <a href="/A248076/b248076.txt">Table of n, a(n) for n = 1..10000</a>
%F A248076 a(n) = Sum_{i=1..n} sigma_5(i) = Sum_{i=1..n} A001160(i).
%F A248076 a(n) ~ Zeta(6) * n^6 / 6. - _Vaclav Kotesovec_, Sep 02 2018
%F A248076 a(n) ~ Pi^6 * n^6 / 5670. - _Vaclav Kotesovec_, Sep 02 2018
%F A248076 a(n) = Sum_{k=1..n} (Bernoulli(6, floor(1 + n/k)) - 1/42)/6, where Bernoulli(n,x) are the Bernoulli polynomials. - _Daniel Suteu_, Nov 07 2018
%F A248076 a(n) = Sum_{k=1..n} k^5 * floor(n/k). - _Daniel Suteu_, Nov 08 2018
%p A248076 with(numtheory): A248076:=n->add(sigma[5](i), i=1..n): seq(A248076(n), n=1..50);
%t A248076 Table[Sum[DivisorSigma[5, i], {i, n}], {n, 30}]
%t A248076 Accumulate[DivisorSigma[5, Range[30]]] (* _Vaclav Kotesovec_, Mar 30 2018 *)
%o A248076 (PARI) lista(nn) = vector(nn, n, sum(i=1, n, sigma(i, 5))) \\ _Michel Marcus_, Sep 30 2014
%o A248076 (Magma) [(&+[DivisorSigma(5,j): j in [1..n]]): n in [1..30]]; // _G. C. Greubel_, Nov 07 2018
%o A248076 (Python)
%o A248076 from math import isqrt
%o A248076 def A248076(n): return ((s:=isqrt(n))**3*(s+1)**2*(1-2*s*(s+1)) + sum((q:=n//k)*(12*k**5+q*(q**2*(q*(2*q+6)+5)-1)) for k in range(1,s+1)))//12 # _Chai Wah Wu_, Oct 21 2023
%Y A248076 Cf. A001160 (sigma_5).
%Y A248076 Cf. A024916: Partial sums of sigma(n)   = A000203(n).
%Y A248076 Cf. A064602: Partial sums of sigma_2(n) = A001157(n).
%Y A248076 Cf. A064603: Partial sums of sigma_3(n) = A001158(n).
%Y A248076 Cf. A064604: Partial sums of sigma_4(n) = A001159(n).
%K A248076 nonn,easy
%O A248076 1,2
%A A248076 _Wesley Ivan Hurt_, Sep 30 2014