The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A060648 Number of cyclic subgroups of the group C_n X C_n (where C_n is the cyclic group of order n). 34
 1, 4, 5, 10, 7, 20, 9, 22, 17, 28, 13, 50, 15, 36, 35, 46, 19, 68, 21, 70, 45, 52, 25, 110, 37, 60, 53, 90, 31, 140, 33, 94, 65, 76, 63, 170, 39, 84, 75, 154, 43, 180, 45, 130, 119, 100, 49, 230, 65, 148, 95, 150, 55, 212, 91, 198, 105, 124, 61, 350, 63, 132, 153, 190 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The group U(n) of units modulo n acts on the direct product (Z_n)^k by multiplication. The number g(n,k) of orbits of U(n) acting on Z/(n)^k is g(n,k) = (1/phi(n))*Sum(gcd(n,a-1)^k) where the sum is over a in U(n) and phi(n) is the Euler totient function. A060648 gives g(n,2). - W. Edwin Clark, Jul 20 2001 a(n) is also the number of orbits of length n for the map TxT (Cartesion product) where T is a map with one orbit of each length. - Thomas Ward, Apr 08 2009 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], 2012-2014. - 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. Apisit Pakapongpun and Thomas Ward, Functorial orbit counting, Journal of Integer Sequences, 12 (2009) Article 09.2.4. László Tóth, Menon's identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110, and arXiv:1103.5861, [math.NT], 2011. László Tóth, On the number of cyclic subgroups of a finite abelian group, arXiv: 1203.6201 [math.GR], 2012. László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013-2014. 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)). If n = p^e, p prime, a(n) = (p^(e+1)+p^e-2)/(p-1). a(n) = Sum_{i|n, j|n} phi(i)*phi(j)/phi(lcm(i, j)). - Vladeta Jovovic, Jul 07 2001 Also a(n) = Sum_{i|n, j|n} phi(gcd(i, j)). Also a(n) = Sum_{d|n} phi(n/d)*tau(d^2). a(n) = sum(d|n, sigma(d)*moebius(n/d)^2 ). - Benoit Cloitre, Sep 08 2002 Inverse Euler transform of A156302. - Vladeta Jovovic, Feb 14 2009 Moebius transform of A060724. - Vladeta Jovovic, Apr 05 2009 Also a(n) = (1/n)*Sum_{d|n} sigma(d)^2*moebius(n/d). - Vladeta Jovovic, Mar 31 2009 Inverse Moebius transform of A001615. - Vladeta Jovovic, Apr 05 2009 From Thomas Ward, Apr 08 2009: (Start) a(n) = Sum_{lcm(e,d)=n} gcd(e,d). Dirichlet g.f.: zeta(s)^2*zeta(s-1)/zeta(2s). (End) For the proofs of these formulas see the papers of L. Toth. a(n) = Sum_{d|n} psi(d), where psi is Dedekind's psi function A001615. - Peter Luschny, Sep 10 2012 a(n) = Sum_{d|n} 2^omega(d)*(n/d). - Peter Luschny, Sep 15 2012 EXAMPLE The cycle index of C_4 X C_4 is (x(1)^4 + x(2)^2 + 2*x(4))^2 = x(1)^8 + 2*x(1)^4*x(2)^2 + 4*x(1)^4*x(4) + x(2)^4 + 4*x(2)^2*x(4) + 4*x(4)^2 and C_4 X C_4 has 1 element of order 1, 3 elements of order 2 and 12 elements of order 4. So a(4) = 1/phi(1) + 3/phi(2) + 12/phi(4) = 10, where phi = Euler totient function, cf. A000010. - Vladeta Jovovic, Jul 05 2001 For a(4) the pairs (e,d) are (1,4),(2,4),(4,4),(4,2),(4,1) with gcds 1,2,4,2,1 resp. giving 10 in total. - Thomas Ward, Apr 08 2009 MAPLE for n from 1 to 200 do:ans := 1:for i from 1 to nops(ifactors(n)) do p := ifactors(n)[i]:e := ifactors(n)[i]:ans := ans*(p^(e+1)+p^e-2)/(p-1):od:printf(`%d, `, ans):od: MATHEMATICA Table[ Plus @@ Map[ Times @@ (EulerPhi /@ #)/EulerPhi[ LCM @@ # ] &, Flatten[ Outer[ {##} &, Divisors[ i ], Divisors[ i ] ], 1 ] ], {i, 1, 100} ] f[p_, e_] := (p^(e+1)+p^e-2)/(p-1); a = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 20 2020 *) PROG (Sage) def A060648(n) :     def dedekind_psi(n) : return n*mul(1+1/p for p in prime_divisors(n))     return reduce(lambda x, y: x+y, [dedekind_psi(d) for d in divisors(n)]) [A060648(n) for n in (1..64)]  # Peter Luschny, Sep 10 2012 (PARI) a(n) = sumdiv(n, d,  2^omega(d)*(n/d) ); \\ Joerg Arndt, Sep 16 2012 CROSSREFS Cf. A060724, A063379, A061503, A216620. Sequence in context: A009778 A215754 A226486 * A263828 A327577 A092961 Adjacent sequences:  A060645 A060646 A060647 * A060649 A060650 A060651 KEYWORD nonn,mult,easy AUTHOR Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 04 2001 EXTENSIONS More terms and additional comments from Vladeta Jovovic, Jul 05 2001 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified July 26 20:56 EDT 2021. Contains 346300 sequences. (Running on oeis4.)