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

%I A065916 #38 Dec 14 2024 20:31:06
%S A065916 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,
%T A065916 31,7,127,7,31,7,8191,7,31,7,127,7,31,7,511,7,31,7,127,7,31,7,2047,7,
%U A065916 31,7,127,7,31,7,511,7,31,7,127,7,31,7,32767,7,31,7,127,7,31,7,511,7
%N A065916 Denominator of sigma(8*n^2)/sigma(4*n^2).
%C A065916 The sequence is not periodic. The denominators are always of the form -1+2^s.
%H A065916 Harry J. Smith, <a href="/A065916/b065916.txt">Table of n, a(n) for n = 1..1000</a>
%H A065916 Ralf Stephan, <a href="/somedcgf.html">Some divide-and-conquer sequences with (relatively) simple ordinary generating functions</a>, 2004.
%H A065916 Ralf Stephan, <a href="/A079944/a079944.ps">Table of generating functions</a>.
%H A065916 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>.
%F A065916 From _Johannes W. Meijer_, Feb 12 2013: (Start)
%F A065916 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).
%F A065916 a(2^(p+3)*n + 2^(p+2) - 1) = a(2^(p+2)*n + 2^(p+1) - 1) for p >= 0. (End)
%F A065916 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
%e A065916 sigma(72)/sigma(36) = 15/7, so a(3) = 7.
%p A065916 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
%t A065916 a[n_] := 2^(2*IntegerExponent[n, 2] + 3) - 1; Array[a, 100] (* _Amiram Eldar_, Jun 21 2024 *)
%o A065916 (PARI) a(n) = denominator(sigma(8*n^2)/sigma(4*n^2)) \\ _Harry J. Smith_, Nov 04 2009
%o A065916 (PARI) a(n)=2^(2*valuation(n,2)+3)-1 \\ _Charles R Greathouse IV_, Nov 18 2015
%Y A065916 Cf. A000203, A007814, A028982, A065915 (numerators), A083420, A220466.
%K A065916 nonn,frac,easy
%O A065916 1,1
%A A065916 _Labos Elemer_, Nov 28 2001