A060724 - OEIS

%I #53 May 11 2021 10:35:25

%S 1,5,6,15,8,30,10,37,23,40,14,90,16,50,48,83,20,115,22,120,60,70,26,

%T 222,45,80,76,150,32,240,34,177,84,100,80,345,40,110,96,296,44,300,46,

%U 210,184,130,50,498,75,225,120,240,56,380,112,370,132,160,62,720,64

%N Number of subgroups of the group C_n X C_n (where C_n is the cyclic group of order n).

%H Enrique Pérez Herrero, <a href="/A060724/b060724.txt">Table of n, a(n) for n = 1..5000</a>

%H M. Hampejs, N. Holighaus, L. Toth and C. Wiesmeyr, <a href="http://arxiv.org/abs/1211.1797">On the subgroups of the group Z_m X Z_n</a>, arXiv preprint arXiv:1211.1797 [math.GR], 2012. - From _N. J. A. Sloane_, Jan 02 2013

%H W. G. Nowak and L. Tóth, <a href="http://arxiv.org/abs/1307.1414">On the average number of subgroups of the group Z_m X Z_n</a>, arXiv preprint arXiv:1307.1414 [math.NT], 2013.

%H Laszlo Toth, <a href="http://arxiv.org/abs/1203.6201">On the number of cyclic subgroups of a finite abelian group</a>, arXiv preprint arXiv:1203.6201 [math.GR], 2012. - From _N. J. A. Sloane_, Sep 22 2012

%H L. Tóth, <a href="http://arxiv.org/abs/1310.7053">Multiplicative arithmetic functions of several variables: a survey</a>, arXiv preprint arXiv:1310.7053 [math.NT], 2013.

%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>

%F a(n) is multiplicative: if the canonical factorization of n is the product of p^e(p) over primes then a(n) = product a(p^e(p)). For a prime p: a(p) = p + 3.

%F a(p^e) = (p^(e+2)+p^(e+1)+1+2*e-3*p-2*e*p)/(p-1)^2.

%F a(n) = Sum_{i|n, j|n} gcd(i, j). - _Vladeta Jovovic_, Oct 28 2001

%F Also a(n) = Sum_{d|n} d*tau((n/d)^2). - _Vladeta Jovovic_, Apr 01 2002

%F Also a(n) = Sum_{d|n} phi(n/d)*tau(d)^2.

%F Inverse Moebius transform of A060648. - _Vladeta Jovovic_, Mar 31 2009

%F Dirichlet g.f. zeta^3(s)*zeta(s-1)/zeta(2*s). - _R. J. Mathar_, Mar 14 2011

%F a(n) = Sum_{d|n} psi(d)*tau(n/d), where psi is A001615 and tau is A000005. - _Enrique Pérez Herrero_, Feb 29 2012

%F Sum_{k=1..n} a(k) ~ 5 * Pi^2 * n^2 / 24. - _Vaclav Kotesovec_, Jun 02 2019

%F a(n) = Sum_{k=1..n} tau(gcd(k,n))^2. - _Seiichi Manyama_, May 11 2021

%e a(2) = 5 because for the group C_2 X C_2 there are the following subgroups: the trivial subgroup, the whole group and the three subgroups of order 2.

%p for n from 1 to 200 do: ans := 1: for i from 1 to nops(ifactors(n)[2]) do p := ifactors(n)[2][i][1]: e := ifactors(n)[2][i][2]: ans := ans*(p^(e+2)+p^(e+1)+1+2*e-3*p-2*e*p)/(p-1)^2: od: printf(`%d,`,ans): od:

%t ppCase[ {p_Integer, e_Integer} ] := (1-2*e*(p-1)+p*(p^e*(1+p)-3))/(p-1)^2; Table[ Times @@ (ppCase /@ FactorInteger[ i ]), {i, 1, 100} ]

%o (GAP) List([1..50], n->Sum(ConjugacyClassesSubgroups( LatticeSubgroups( DirectProduct( List([n, n], k->CyclicGroup(k)) ))), Size)); # _Andrew Howroyd_, Jul 01 2018

%o (PARI) a(n)={sumdiv(n, d, eulerphi(n/d)*numdiv(d)^2)} \\ _Andrew Howroyd_, Jul 01 2018

%o (PARI) a(n) = sum(k=1, n, numdiv(gcd(k, n))^2); \\ _Seiichi Manyama_, May 11 2021

%o (Sage)

%o def A060724(n) :

%o     d = divisors(n); cp = cartesian_product([d, d])

%o     return reduce(lambda x,y: x+y, map(gcd, cp))

%o [A060724(n) for n in (1..61)]   # _Peter Luschny_, Sep 10 2012

%Y Cf. A060648, A050488, A054584, A000005, A062369, A344132, A344138, A344139, A344140.

%Y Main diagonal of A216624.

%K nonn,mult

%O 1,2

%A Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001

%E Formula and more terms from _Vladeta Jovovic_, Jul 06 2001