A017689 Numerator of sum of -13th powers of divisors of n.

1, 8193, 1594324, 67117057, 1220703126, 1088524711, 96889010408, 549822930945, 2541867422653, 5000610355659, 34522712143932, 26751583696117, 302875106592254, 99226457784093, 648732096885608, 4504149450301441, 9904578032905938, 6941839931265343, 42052983462257060

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• frac

Comments

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A017690 (denominator), A013671, A013672.

Programs

Magma

[Numerator(DivisorSigma(13,n)/n^13): n in [1..20]]; // G. C. Greubel, Nov 06 2018

Maple

A017689 := proc(n)
    numtheory[sigma][-13](n) ;
    numer(%) ;
end proc: # R. J. Mathar, Sep 21 2017

Mathematica

Table[Numerator[DivisorSigma[13, n]/n^13], {n, 1, 20}] (* G. C. Greubel, Nov 06 2018 *)

PARI

vector(20, n, numerator(sigma(n, 13)/n^13)) \\ G. C. Greubel, Nov 06 2018

Formula

From Amiram Eldar, Apr 02 2024: (Start)

sup_{n>=1} a(n)/A017690(n) = zeta(13) (A013671).

Dirichlet g.f. of a(n)/A017690(n): zeta(s)*zeta(s+13).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017690(k) = zeta(14) (A013672). (End)