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

1, 34, 278, 1335, 4461, 12513, 29321, 63146, 122439, 225597, 386649, 644557, 1015851, 1570515, 2333259, 3415660, 4835518, 6792187, 9268287, 12572469, 16673621, 21988337, 28424681, 36677981, 46446732, 58699434, 73107634, 90873690, 111384840, 136555392

Offset: 1

Author

Wesley Ivan Hurt, Sep 30 2014

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001160 (sigma_5).

Cf. A024916: Partial sums of sigma(n) = A000203(n).

Cf. A064602: Partial sums of sigma_2(n) = A001157(n).

Cf. A064603: Partial sums of sigma_3(n) = A001158(n).

Cf. A064604: Partial sums of sigma_4(n) = A001159(n).

Programs

Magma

[(&+[DivisorSigma(5,j): j in [1..n]]): n in [1..30]]; // G. C. Greubel, Nov 07 2018

Maple

with(numtheory): A248076:=n->add(sigma[5](i), i=1..n): seq(A248076(n), n=1..50);

Mathematica

Table[Sum[DivisorSigma[5, i], {i, n}], {n, 30}]
Accumulate[DivisorSigma[5, Range[30]]] (* Vaclav Kotesovec, Mar 30 2018 *)

PARI

lista(nn) = vector(nn, n, sum(i=1, n, sigma(i, 5))) \\ Michel Marcus, Sep 30 2014

Python

from math import isqrt
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

Formula

a(n) = Sum_{i=1..n} sigma_5(i) = Sum_{i=1..n} A001160(i).

a(n) ~ Zeta(6) * n^6 / 6. - Vaclav Kotesovec, Sep 02 2018

a(n) ~ Pi^6 * n^6 / 5670. - Vaclav Kotesovec, Sep 02 2018

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

a(n) = Sum_{k=1..n} k^5 * floor(n/k). - Daniel Suteu, Nov 08 2018