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.
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) a(n) = denominator(sigma(8*n^2)/sigma(4*n^2)) \\ Harry J. Smith, Nov 04 2009
(PARI) a(n)=2^(2*valuation(n, 2)+3)-1 \\ Charles R Greathouse IV, Nov 18 2015
CROSSREFS
KEYWORD
nonn,frac,easy,changed
AUTHOR
Labos Elemer, Nov 28 2001
STATUS
approved