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A064767 Order of automorphism group of the group C_n X C_n X C_n (where C_n is the cyclic group of order n). 14
1, 168, 11232, 86016, 1488000, 1886976, 33784128, 44040192, 221079456, 249984000, 2124276000, 966131712, 9726417792, 5675733504, 16713216000, 22548578304, 111203278848, 37141348608, 304812862560, 127991808000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Also number of 3 X 3 invertible matrices over the ring Z/nZ. - Max Alekseyev, Nov 02 2007

Order of the group GL(3,Z_n). For n > 2, a(n) is divisible by 96. - Jianing Song, Nov 24 2018

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.

C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, arXiv:math/0605185 [math.GR], 2006.

C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, Amer. Math. Monthly 114 (2007), no 10, 917-923.

J. Overbey, W. Traves and J. Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29 , Iss. 1, 2005.

Index entries for sequences related to groups

FORMULA

a(n) = phi(n)*A011785(n). - Vladeta Jovovic, Oct 29 2001

a(n) = n^9*Product_{primes p dividing n} ((1 - 1/p^3)*(1 - 1/p^2)*(1 - 1/p)). This also gives a formula for A011785.

Multiplicative with a(p^e) = p^(9*e-6)*(p^3 - 1)*(p^2 - 1)*(p - 1). - Vladeta Jovovic, Nov 18 2001

MATHEMATICA

a[n_] := n^9*Times @@ Function[p, (1 - 1/p^3)*(1 - 1/p^2)*(1 - 1/p)] /@ FactorInteger[n][[All, 1]]; a[1] = 1; Array[a, 20] (* Jean-Fran├žois Alcover, Mar 21 2017 *)

PROG

(PARI) a(n) = n^9*prod(k=2, n, if (!isprime(k) || (n % k), 1, (1-1/k^3)*(1-1/k^2)*(1-1/k))); \\ Michel Marcus, Jun 30 2015

CROSSREFS

Row n=3 of A316622.

Cf. A064803.

Cf. A000252 (GL(2,Z_n)), A305186 (GL(4,Z_n)).

Cf. A000056 (SL(2,Z_n)), A011785 (SL(3,Z_n)), A011786 (SL(4,Z_n)).

Sequence in context: A271033 A019286 A218413 * A115222 A290152 A282586

Adjacent sequences:  A064764 A064765 A064766 * A064768 A064769 A064770

KEYWORD

nonn,easy,mult

AUTHOR

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Oct 24 2001

EXTENSIONS

More terms from Vladeta Jovovic, Nov 18 2001

STATUS

approved

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Last modified October 15 01:40 EDT 2019. Contains 328025 sequences. (Running on oeis4.)