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

1, 18, 100, 373, 999, 2393, 4795, 9164, 15807, 26449, 41091, 63477, 92039, 132873, 184205, 254110, 337632, 450563, 580885, 751783, 948747, 1197661, 1477503, 1835761, 2227012, 2712566, 3250650, 3906396, 4613678, 5486322, 6409844

Offset

1,2

Comments

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

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = a(n-1) + A001159(n) = Sum_{j=1..n} sigma_4(j), where sigma_4(j) = A001159(j).

G.f.: (1/(1 - x))*Sum_{k>=1} k^4*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 23 2017

a(n) ~ zeta(5) * n^5 / 5. - Vaclav Kotesovec, Sep 02 2018

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

a(n) = Sum_{k=1..n} k^4 * floor(n/k). - Daniel Suteu, Nov 08 2018

Maple

ListTools:-PartialSums(map(numtheory:-sigma[4], [$1..100])); # Robert Israel, Jun 29 2018

Mathematica

Accumulate[DivisorSigma[4, Range[50]]] (* Vaclav Kotesovec, Mar 30 2018 *)

Prog

(PARI) vector(50, n, sum(j=1, n, sigma(j, 4))) \\ G. C. Greubel, Nov 07 2018
(Magma) [(&+[DivisorSigma(4, j): j in [1..n]]): n in [1..50]]; // G. C. Greubel, Nov 07 2018
(Python)
from math import isqrt
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

Crossrefs

Cf. A001159, A064607.

Sequence in context: A231144 A259231 A376877 * A359435 A301542 A231138

Adjacent sequences: A064601 A064602 A064603 * A064605 A064606 A064607

Keyword

nonn

Author

Labos Elemer, Sep 24 2001

Status

approved