OFFSET
1,1
COMMENTS
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
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..16384
Jon Maiga, Efficient computation of ratios between divisor sums, 2018.
FORMULA
a((2*n-1)*2^p) = 2^(p+2)-1, p >= 0 and n >= 1. - Johannes W. Meijer, Feb 09 2013
a(n) = (2n) XOR (2n-1). - Jon Maiga, Dec 10 2018
From Amiram Eldar, Jan 06 2023: (Start)
Sum_{k=1..n} a(k) ~ (log_2(n) + (gamma-1)/log(2) + 1)*2*n, where gamma is Euler's constant (A001620).
MAPLE
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
MATHEMATICA
k=2; Table[Numerator[DivisorSigma[1, k*n]/DivisorSigma[1, n]], {n, 1, 128}]
Table[BitXor[2*n, 2*n - 1], {n, 128}] (* Jon Maiga, Dec 10 2018 *)
PROG
(PARI) A088837(n) = numerator(sigma(n<<1)/sigma(n)); \\ Antti Karttunen, Nov 01 2018
CROSSREFS
KEYWORD
easy,nonn,frac
AUTHOR
Labos Elemer, Nov 04 2003
STATUS
approved