A017695 Numerator of sum of -16th powers of divisors of n.

1, 65537, 43046722, 4295032833, 152587890626, 1410576509857, 33232930569602, 281479271743489, 1853020231898563, 5000076293978081, 45949729863572162, 30814514057170571, 665416609183179842, 1088993285370003137, 6568408508343827972, 18447025552981295105, 48661191875666868482

Offset

1,2

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

Formula

From Amiram Eldar, Apr 02 2024: (Start)

sup_{n>=1} a(n)/A017696(n) = zeta(16) (A013674).

Dirichlet g.f. of a(n)/A017696(n): zeta(s)*zeta(s+16).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017696(k) = zeta(17) (A013675). (End)

Mathematica

Table[Numerator[Total[1/Divisors[n]^16]], {n, 20}] (* Harvey P. Dale, Sep 26 2014 *)
Table[Numerator[DivisorSigma[16, n]/n^16], {n, 1, 20}] (* G. C. Greubel, Nov 05 2018 *)

Prog

(PARI) vector(20, n, numerator(sigma(n, 16)/n^16)) \\ G. C. Greubel, Nov 05 2018
(Magma) [Numerator(DivisorSigma(16, n)/n^16): n in [1..20]]; // G. C. Greubel, Nov 05 2018

Crossrefs

Cf. A017696 (denominator), A013674, A013675.

Sequence in context: A123388 A070816 A133864 * A013964 A036094 A133865

Adjacent sequences: A017692 A017693 A017694 * A017696 A017697 A017698

Keyword

nonn,frac

Author

N. J. A. Sloane

Status

approved