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A088838
Numerator of the quotient sigma(3n)/sigma(n).
8
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
1,1
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)
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}]
PROG
(PARI) a(n) = numerator(sigma(3*n)/sigma(n)) \\ Felix Fröhlich, Nov 19 2017
CROSSREFS
KEYWORD
easy,nonn,frac,look
AUTHOR
Labos Elemer, Nov 04 2003
STATUS
approved