A088837 Numerator of sigma(2*n)/sigma(n). Denominator see in A038712.

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, 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, 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

Offset: 1

Author

Labos Elemer, Nov 04 2003

Keywords

• easy
• nonn
• frac

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.
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A000203, A038712 (denominators), A062731, A088838, A088839, A088840, A080278, A220466.

Cf. A001620, A065442.

Programs

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 *)

PARI

A088837(n) = numerator(sigma(n<<1)/sigma(n)); \\ Antti Karttunen, Nov 01 2018

Formula

a(n) = 4*2^A007814(n)-1 = 4*A006519(n)-1 = A059159(n)-1 = 2*A038712(n) + 1.

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)

a(n) = numerator(A062731(n)/A000203(n)).

Sum_{k=1..n} a(k) ~ (log_2(n) + (gamma-1)/log(2) + 1)*2*n, where gamma is Euler's constant (A001620).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A038712(k) = A065442 + 1 = 2.606695... . (End).