cp's OEIS Frontend

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

%I A088838 #34 Jan 06 2023 06:33:11
%S A088838 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,
%T A088838 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,
%U A088838 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
%N A088838 Numerator of the quotient sigma(3n)/sigma(n).
%H A088838 Robert Israel, <a href="/A088838/b088838.txt">Table of n, a(n) for n = 1..10000</a>
%H A088838 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A088838 From _Robert Israel_, Nov 19 2017: (Start)
%F A088838 a(n) = (3^(2+A007949(n))-1)/2.
%F A088838 G.f.: Sum_{k>=0} (3^(k+2)-1)*(x^(3^k)+x^(2*3^k))/(2*(1-x^(3^(k+1)))). (End)
%F A088838 a(n) = sigma(3*n)/(sigma(3*n) - 3*sigma(n)), where sigma(n) = A000203(n). - _Peter Bala_, Jun 10 2022
%F A088838 From _Amiram Eldar_, Jan 06 2023: (Start)
%F A088838 a(n) = numerator(A144613(n)/A000203(n)).
%F A088838 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).
%F A088838 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A080278(k) = 4*A214369 + 1 = 3.728614... . (End)
%p A088838 A088838 := proc(n)
%p A088838     numtheory[sigma](3*n)/numtheory[sigma](n) ;
%p A088838     numer(%) ;
%p A088838 end proc:
%p A088838 seq(A088838(n),n=1..100) ; # _R. J. Mathar_, Nov 19 2017
%p A088838 seq((3^(2+padic:-ordp(n,3))-1)/2, n=1..100); # _Robert Israel_, Nov 19 2017
%t A088838 k=3; Table[Numerator[DivisorSigma[1, k*n]/DivisorSigma[1, n]], {n, 1, 128}]
%o A088838 (PARI) a(n) = numerator(sigma(3*n)/sigma(n)) \\ _Felix Fröhlich_, Nov 19 2017
%Y A088838 Cf. A000203, A007949, A038712, A088837, A088839, A088840, A080278 (denominator), A144613.
%Y A088838 Cf. A001620, A214369.
%K A088838 easy,nonn,frac,look
%O A088838 1,1
%A A088838 _Labos Elemer_, Nov 04 2003