cp's OEIS Frontend

A208133 Total number of subgroups of rank <= 2 of a certain class of abelian groups of order n defined as direct products of Z/(mZ) X Z/(kZ) with m|k.

%I A208133 #33 Jun 18 2020 04:41:23
%S A208133 1,2,2,8,2,4,2,12,9,4,2,16,2,4,4,31,2,18,2,16,4,4,2,24,11,4,14,16,2,8,
%T A208133 2,42,4,4,4,72,2,4,4,24,2,8,2,16,18,4,2,62,13,22,4,16,2,28,4,24,4,4,2,
%U A208133 32,2,4,18,90,4,8,2,16,4,8,2,108,2,4,22,16
%N A208133 Total number of subgroups of rank <= 2 of a certain class of abelian groups of order n defined as direct products of Z/(mZ) X Z/(kZ) with m|k.
%C A208133 Level function l_tau^2(n) of Bhowmik and Wu.
%C A208133 Records occur at 1, 2, 4, 8, 12, 16, 32, 36, 64, 72, 108, 128, 144, 288, 432, 576, 1152, 1296, 2304, 3600, 5184, 7200, 9216, 10368, 14112, 14400, 20736, 28224, 28800, 32400, 57600, ... and they are: 1, 2, 8, 12, 16, 31, 42, 72, 90, 108, 112, 116, 279, 378, 434, 810, 1044, 1302, 2025, 3069, 3780, 4158, 4644, 4872, 4914, 8910, 9450, 10530, 11484, 14322, 22275, ... - _Antti Karttunen_, Mar 21 2018
%D A208133 A. Laurincikas, The universality of Dirichlet series attached to finite Abelian groups, in "Number Theory", Proc. Turku Sympos. on Number Theory, May 31-June 4, 1999, p 179.
%H A208133 Antti Karttunen, <a href="/A208133/b208133.txt">Table of n, a(n) for n = 1..65537</a>
%H A208133 G. Bhowmik, Jie Wu, <a href="http://dx.doi.org/10.1515/crll.2001.004">Zeta function of subgroups of abelian groups and average orders</a>, J. reine angew. Math. 530 (2001) 1-15.
%H A208133 Vaclav Kotesovec, <a href="/A208133/a208133.jpg">Graph - the asymptotic ratio (1000000 terms)</a>
%F A208133 Dirichlet g.f.: zeta(s)^2*zeta(2s)^2*zeta(2s-1)*Product_{primes p} (1 + 1/p^(2s) - 2/p^(3s)).
%F A208133 Sum_{k=1..n} a(k) ~ c * Pi^4 * log(n)^2 * n / 144, where c = A330594 = Product_{primes p} (1 + 1/p^2 - 2/p^3) = 1.10696011195321767665117913000743959294954883365812241904313404497877733241... - _Vaclav Kotesovec_, Dec 18 2019
%F A208133 More precise asymptotics: Let f(s) = Product_{primes p} (1 + 1/p^(2*s) - 2/p^(3*s)), then Sum_{k=1..n} a(k) ~ n*Pi^2 * (Pi^2 * f(1) * log(n)^2 + 2*Pi^2 * log(n) * (f(1) * (-1 + 8*gamma - 48*log(A) + 4*log(2*Pi)) + f'(1)) + Pi^2 * (2*f(1)*(1 + 25*gamma^2 + 576*log(A)^2 + log(A) * (48 - 96*log(2*Pi)) - 8*gamma * (1 + 36*log(A) - 3*log(2*Pi)) - 4*log(2*Pi) + 4*log(2*Pi)^2 - 6*sg1) + 2*(-1 + 8*gamma - 48*log(A) + 4*log(2*Pi))*f'(1) + f''(1)) + 48*f(1)*zeta''(2)) / 144, where f(1) = A330594, f'(1) = A330594 * (-A335705) = f(1) * Sum_{primes p} = -2*(p-3) * log(p) / (p^3 + p - 2) = -0.087825458097278818094375273108270679512035928574..., f''(1) = A330594 * (A335705^2 + A335706) = f'(1)^2/f(1) + f(1) * Sum_{primes p} = 2*p*(2*p^3 - 9*p^2 - 1) * log(p)^2) / (p^3 + p - 2)^2 =  0.26722508718782634450711076996710402451611235402675360769..., zeta''(2) = A201994, A is the Glaisher-Kinkelin constant A074962, gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - _Vaclav Kotesovec_, Jun 18 2020
%p A208133 L300828 := [ 1, 0, 0, 1, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0
%p A208133 ] ;
%p A208133 L010052 := [ 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ];
%p A208133 L037213 := [ 1, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ;
%p A208133 Lx := DIRICHLET(L300828,L037213) ;
%p A208133 Lx := DIRICHLET(Lx,L010052) ;
%p A208133 Lx := DIRICHLET(Lx,L010052) ;
%p A208133 Lx := MOBIUSi(Lx) ;
%p A208133 Lx := MOBIUSi(Lx) ;
%p A208133 # Name of initial list L1 changed to L300828 to refer to sequence A300828 by _Antti Karttunen_, Mar 21 2018
%o A208133 (PARI)
%o A208133 A037213(n) = if(issquare(n),sqrtint(n),0);
%o A208133 A300828(n) = { if(1==n, return(1)); my(val=1, v=factor(n), d=matsize(v)[1]); for(i=1,d, if(v[i,2] < 2 || v[i,2] > 3, return(0)); if (v[i,2] == 3, val *= -2)); return(val); };
%o A208133 a208133s1(n) = sumdiv(n,d,A037213(n/d)*A300828(d));
%o A208133 a208133s2(n) = sumdiv(n,d,issquare(n/d)*a208133s1(d));
%o A208133 a208133s3(n) = sumdiv(n,d,issquare(n/d)*a208133s2(d));
%o A208133 a208133s4(n) = sumdiv(n,d,a208133s3(d));
%o A208133 A208133(n) = sumdiv(n,d,a208133s4(d)); \\ _Antti Karttunen_, Mar 21 2018, after _R. J. Mathar_'s Maple code
%o A208133 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X + 2*X^2)/(1 - X)^3/(1 + X)^2/(1 - p*X^2))[n], ", ")) \\ _Vaclav Kotesovec_, Jun 18 2020
%Y A208133 Cf. A010052, A037213, A300828.
%K A208133 nonn,mult
%O A208133 1,2
%A A208133 _R. J. Mathar_, Mar 29 2012