A060724 Number of subgroups of the group C_n X C_n (where C_n is the cyclic group of order n).
1, 5, 6, 15, 8, 30, 10, 37, 23, 40, 14, 90, 16, 50, 48, 83, 20, 115, 22, 120, 60, 70, 26, 222, 45, 80, 76, 150, 32, 240, 34, 177, 84, 100, 80, 345, 40, 110, 96, 296, 44, 300, 46, 210, 184, 130, 50, 498, 75, 225, 120, 240, 56, 380, 112, 370, 132, 160, 62, 720, 64
Offset: 1
Examples
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.
Links
- Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
- M. Hampejs, N. Holighaus, L. Toth and C. Wiesmeyr, On the subgroups of the group Z_m X Z_n, arXiv preprint arXiv:1211.1797 [math.GR], 2012. - From _N. J. A. Sloane_, Jan 02 2013
- W. G. Nowak and L. Tóth, On the average number of subgroups of the group Z_m X Z_n, arXiv preprint arXiv:1307.1414 [math.NT], 2013.
- Laszlo Toth, On the number of cyclic subgroups of a finite abelian group, arXiv preprint arXiv:1203.6201 [math.GR], 2012. - From _N. J. A. Sloane_, Sep 22 2012
- L. Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013.
- Index entries for sequences related to groups
Crossrefs
Programs
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GAP
List([1..50], n->Sum(ConjugacyClassesSubgroups( LatticeSubgroups( DirectProduct( List([n, n], k->CyclicGroup(k)) ))), Size)); # Andrew Howroyd, Jul 01 2018
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Maple
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:
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Mathematica
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} ] -
PARI
a(n)={sumdiv(n, d, eulerphi(n/d)*numdiv(d)^2)} \\ Andrew Howroyd, Jul 01 2018 -
PARI
a(n) = sum(k=1, n, numdiv(gcd(k, n))^2); \\ Seiichi Manyama, May 11 2021
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Sage
def A060724(n) : d = divisors(n); cp = cartesian_product([d, d]) return reduce(lambda x,y: x+y, map(gcd, cp)) [A060724(n) for n in (1..61)] # Peter Luschny, Sep 10 2012
Formula
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.
a(p^e) = (p^(e+2)+p^(e+1)+1+2*e-3*p-2*e*p)/(p-1)^2.
a(n) = Sum_{i|n, j|n} gcd(i, j). - Vladeta Jovovic, Oct 28 2001
Also a(n) = Sum_{d|n} d*tau((n/d)^2). - Vladeta Jovovic, Apr 01 2002
Also a(n) = Sum_{d|n} phi(n/d)*tau(d)^2.
Inverse Moebius transform of A060648. - Vladeta Jovovic, Mar 31 2009
Dirichlet g.f. zeta^3(s)*zeta(s-1)/zeta(2*s). - R. J. Mathar, Mar 14 2011
a(n) = Sum_{d|n} psi(d)*tau(n/d), where psi is A001615 and tau is A000005. - Enrique Pérez Herrero, Feb 29 2012
Sum_{k=1..n} a(k) ~ 5 * Pi^2 * n^2 / 24. - Vaclav Kotesovec, Jun 02 2019
a(n) = Sum_{k=1..n} tau(gcd(k,n))^2. - Seiichi Manyama, May 11 2021
Extensions
Formula and more terms from Vladeta Jovovic, Jul 06 2001