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.
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
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