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A060724 Number of subgroups of the group C_n X C_n (where C_n is the cyclic group of order n). 16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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

EXAMPLE

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.

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:

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} ]

PROG

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

(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

(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

CROSSREFS

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

Main diagonal of A216624.

Sequence in context: A289190 A145491 A282466 * A064949 A160109 A351172

Adjacent sequences:  A060721 A060722 A060723 * A060725 A060726 A060727

KEYWORD

nonn,mult

AUTHOR

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

EXTENSIONS

Formula and more terms from Vladeta Jovovic, Jul 06 2001

STATUS

approved

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Last modified October 6 12:00 EDT 2022. Contains 357264 sequences. (Running on oeis4.)