A065915 Numerator of sigma(8*n^2)/sigma(4*n^2).

15, 63, 15, 255, 15, 63, 15, 1023, 15, 63, 15, 255, 15, 63, 15, 4095, 15, 63, 15, 255, 15, 63, 15, 1023, 15, 63, 15, 255, 15, 63, 15, 16383, 15, 63, 15, 255, 15, 63, 15, 1023, 15, 63, 15, 255, 15, 63, 15, 4095, 15, 63, 15, 255, 15, 63, 15, 1023, 15, 63, 15, 255, 15, 63

Offset

1,1

Comments

The sequence is not periodic, values of numerators are always -1+2^s.

Links

Harry J. Smith, Table of n, a(n) for n = 1..1000

Formula

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A065916(n) = Sum_{k>=0} (2^(2*k+4)-1)/(2^(k+1)*(2^(2*k + 3)-1)) = 2.080617095034... . - Amiram Eldar, Apr 04 2024

Example

a(3) = sigma(72)/sigma(36) = 15/7.
Fractions begin with 15/7, 63/31, 15/7, 255/127, 15/7, 63/31, 15/7, 1023/511, 15/7, 63/31, 15/7, 255/127, ...

Mathematica

Table[Numerator[DivisorSigma[1, 8n^2]/DivisorSigma[1, 4n^2]], {n, 70}] (* Harvey P. Dale, Mar 21 2018 *)

Prog

(PARI) for (n=1, 1000, a=numerator(sigma(8*n^2)/sigma(4*n^2)); write("b065915.txt", n, " ", a)) \\ Harry J. Smith, Nov 04 2009
(PARI) a(n)=2^(2*valuation(n, 2)+4)-1 \\ Charles R Greathouse IV, Nov 17 2015

Crossrefs

Cf. A000203, A065916 (denominator).

Sequence in context: A219296 A067572 A066584 * A062965 A157968 A141759

Adjacent sequences: A065912 A065913 A065914 * A065916 A065917 A065918

Keyword

nonn,easy,frac

Author

Labos Elemer, Nov 28 2001

Status

approved