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

1,1

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.

Index entries for sequences related to sigma(n).

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).

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

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

Cf. A001620, A065442.

Sequence in context: A316255 A096385 A205723 * A201385 A186107 A282160

Adjacent sequences: A088834 A088835 A088836 * A088838 A088839 A088840

Keyword

easy,nonn,frac

Author

Labos Elemer, Nov 04 2003

Status

approved