A065916 Denominator of sigma(8*n^2)/sigma(4*n^2).

7, 31, 7, 127, 7, 31, 7, 511, 7, 31, 7, 127, 7, 31, 7, 2047, 7, 31, 7, 127, 7, 31, 7, 511, 7, 31, 7, 127, 7, 31, 7, 8191, 7, 31, 7, 127, 7, 31, 7, 511, 7, 31, 7, 127, 7, 31, 7, 2047, 7, 31, 7, 127, 7, 31, 7, 511, 7, 31, 7, 127, 7, 31, 7, 32767, 7, 31, 7, 127, 7, 31, 7, 511, 7

Offset

1,1

Comments

The sequence is not periodic. The denominators are always of the form -1+2^s.

Links

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

Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.

Ralf Stephan, Table of generating functions.

Index entries for sequences related to binary expansion of n.

Formula

From Johannes W. Meijer, Feb 12 2013: (Start)

a((2*n-1)*2^p) = 2*4^(p+1) - 1 for p >= 0 and n >= 1. Observe that a(2^p) = A083420(p+1).

a(2^(p+3)*n + 2^(p+2) - 1) = a(2^(p+2)*n + 2^(p+1) - 1) for p >= 0. (End)

a(n) = 2^s-1, with s = 2*A007814(n) + 3. Recurrence: a(2n) = 4a(n)+3, a(2n+1) = 7. - Ralf Stephan, Aug 22 2013

Example

sigma(72)/sigma(36) = 15/7, so a(3) = 7.

Maple

nmax:=73: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := 2*4^(p+1) - 1 od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Feb 12 2013

Mathematica

a[n_] := 2^(2*IntegerExponent[n, 2] + 3) - 1; Array[a, 100] (* Amiram Eldar, Jun 21 2024 *)

Prog

(PARI) for (n=1, 1000, a=denominator(sigma(8*n^2)/sigma(4*n^2)); write("b065916.txt", n, " ", a) ) \\ Harry J. Smith, Nov 04 2009
(PARI) a(n)=2^(2*valuation(n, 2)+3)-1 \\ Charles R Greathouse IV, Nov 18 2015

Crossrefs

Cf. A000203, A007814, A028982, A065915 (numerators), A083420, A220466.

Sequence in context: A196315 A156347 A221402 * A122654 A184121 A167768

Adjacent sequences: A065913 A065914 A065915 * A065917 A065918 A065919

Keyword

nonn,easy

Author

Labos Elemer, Nov 28 2001

Status

approved