A088838 Numerator of the quotient sigma(3n)/sigma(n).

4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 121, 4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 121, 4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 364, 4, 4, 13, 4, 4, 13, 4, 4, 40

Offset: 1

Author

Labos Elemer, Nov 04 2003

Keywords

• easy
• nonn
• frac
• look

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A000203, A007949, A038712, A088837, A088839, A088840, A080278 (denominator), A144613.

Cf. A001620, A214369.

Programs

Maple

A088838 := proc(n)
    numtheory[sigma](3*n)/numtheory[sigma](n) ;
    numer(%) ;
end proc:
seq(A088838(n),n=1..100) ; # R. J. Mathar, Nov 19 2017
seq((3^(2+padic:-ordp(n,3))-1)/2, n=1..100); # Robert Israel, Nov 19 2017

Mathematica

k=3; Table[Numerator[DivisorSigma[1, k*n]/DivisorSigma[1, n]], {n, 1, 128}]

PARI

a(n) = numerator(sigma(3*n)/sigma(n)) \\ Felix Fröhlich, Nov 19 2017

Formula

From Robert Israel, Nov 19 2017: (Start)

a(n) = (3^(2+A007949(n))-1)/2.

G.f.: Sum_{k>=0} (3^(k+2)-1)*(x^(3^k)+x^(2*3^k))/(2*(1-x^(3^(k+1)))). (End)

a(n) = sigma(3*n)/(sigma(3*n) - 3*sigma(n)), where sigma(n) = A000203(n). - Peter Bala, Jun 10 2022

From Amiram Eldar, Jan 06 2023: (Start)

a(n) = numerator(A144613(n)/A000203(n)).

Sum_{k=1..n} a(k) ~ (3/log(3))*n*log(n) + (1/2 + 3*(gamma-1)/log(3))*n, where gamma is Euler's constant (A001620).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A080278(k) = 4*A214369 + 1 = 3.728614... . (End)