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

Author

Labos Elemer, Nov 28 2001

Keywords

• nonn
• easy
• frac

Comments

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

Examples

			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, ...
		

Links

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

Crossrefs

Cf. A000203, A065916 (denominator).

Programs

Magma

A065915:= func< n | 2^(2*Valuation(n, 2)+4) -1 >;
[A065915(n): n in [1..100]]; // G. C. Greubel, Aug 25 2024

Mathematica

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

PARI

a(n) = numerator(sigma(8*n^2)/sigma(4*n^2)) \\ Harry J. Smith, Nov 04 2009

PARI

a(n)=2^(2*valuation(n,2)+4)-1 \\ Charles R Greathouse IV, Nov 17 2015

SageMath

def A065915(n): return 2^(2*valuation(n, 2)+4) -1
[A065915(n) for n in range(1,101)] # G. C. Greubel, Aug 25 2024

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