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%I #33 Jan 06 2023 06:32:50
%S 3,7,3,15,3,7,3,31,3,7,3,15,3,7,3,63,3,7,3,15,3,7,3,31,3,7,3,15,3,7,3,
%T 127,3,7,3,15,3,7,3,31,3,7,3,15,3,7,3,63,3,7,3,15,3,7,3,31,3,7,3,15,3,
%U 7,3,255,3,7,3,15,3,7,3,31,3,7,3,15,3,7,3,63,3,7,3,15,3,7,3,31,3,7,3,15,3
%N Numerator of sigma(2*n)/sigma(n). Denominator see in A038712.
%C In general sigma(2^k*n) / sigma(n) = ((2^k*n) XOR (2^k*n-1)) / (n XOR (n-1)), see link. _Jon Maiga_, Dec 10 2018
%H Antti Karttunen, <a href="/A088837/b088837.txt">Table of n, a(n) for n = 1..16384</a>
%H Jon Maiga, <a href="http://jonkagstrom.com/divisor-sum-bitwise/">Efficient computation of ratios between divisor sums</a>, 2018.
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F a(n) = 4*2^A007814(n)-1 = 4*A006519(n)-1 = A059159(n)-1 = 2*A038712(n) + 1.
%F a((2*n-1)*2^p) = 2^(p+2)-1, p >= 0 and n >= 1. - _Johannes W. Meijer_, Feb 09 2013
%F a(n) = (2n) XOR (2n-1). - _Jon Maiga_, Dec 10 2018
%F From _Amiram Eldar_, Jan 06 2023: (Start)
%F a(n) = numerator(A062731(n)/A000203(n)).
%F Sum_{k=1..n} a(k) ~ (log_2(n) + (gamma-1)/log(2) + 1)*2*n, where gamma is Euler's constant (A001620).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A038712(k) = A065442 + 1 = 2.606695... . (End).
%p nmax:=93: 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^(p+2)-1 od: od: seq(a(n), n=1..nmax); # _Johannes W. Meijer_, Feb 09 2013
%t k=2; Table[Numerator[DivisorSigma[1, k*n]/DivisorSigma[1, n]], {n, 1, 128}]
%t Table[BitXor[2*n, 2*n - 1], {n, 128}] (* _Jon Maiga_, Dec 10 2018 *)
%o (PARI) A088837(n) = numerator(sigma(n<<1)/sigma(n)); \\ _Antti Karttunen_, Nov 01 2018
%Y Cf. A000203, A038712 (denominators), A062731, A088838, A088839, A088840, A080278, A220466.
%Y Cf. A001620, A065442.
%K easy,nonn,frac
%O 1,1
%A _Labos Elemer_, Nov 04 2003
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