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A064604 Partial sums of A001159: Sum_{j=1..n} sigma_4(j). 9
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 (list; graph; refs; listen; history; text; internal format)
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

CROSSREFS

Cf. A001159, A064607.

Sequence in context: A087638 A231144 A259231 * A301542 A231138 A140198

Adjacent sequences:  A064601 A064602 A064603 * A064605 A064606 A064607

KEYWORD

nonn

AUTHOR

Labos Elemer, Sep 24 2001

STATUS

approved

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Last modified July 23 11:28 EDT 2021. Contains 346259 sequences. (Running on oeis4.)