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A305186 Number of invertible 4 X 4 matrices mod n. 7
1, 20160, 24261120, 1321205760, 116064000000, 489104179200, 27811094169600, 86586540687360, 1044361663787520, 2339850240000000, 41393302251840000, 32053931488051200, 610296923230525440 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Order of the group GL(4,Z_n).

Order of the automorphism group of the group (C_n)^4, where C_n is the cyclic group of order n.

For n > 2, a(n) is divisible by 23040.

LINKS

Jianing Song, Table of n, a(n) for n = 1..10000

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.

FORMULA

Multiplicative with a(p^e) = (p - 1)*(p^2 - 1)*(p^3 - 1)*(p^4 - 1)*p^(16*e-10).

a(n) = n^16*Product_{primes p dividing n} (1 - 1/p^4)*(1 - 1/p^3)*(1 - 1/p^2)*(1 - 1/p).

a(n) = phi(n)*A011786(n) = A000010(n)*A011786(n).

MATHEMATICA

{1}~Join~Array[#^16*Product[(1 - 1/p^4) (1 - 1/p^3) (1 - 1/p^2) (1 - 1/p), {p, FactorInteger[#][[All, 1]]}] &, 12, 2] (* Michael De Vlieger, May 27 2018 *)

PROG

(PARI) a(n)=my(f=factor(n)[, 1]); n^16*prod(i=1, #f, (1-1/f[i]^4)*(1-1/f[i]^3)*(1-1/f[i]^2)*(1-1/f[i]))

CROSSREFS

Row n=4 of A316622.

Cf. A000252 (GL(2,Z_n)), A064767 (GL(3,Z_n)).

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

Sequence in context: A003808 A011786 A003801 * A181233 A262776 A119648

Adjacent sequences:  A305183 A305184 A305185 * A305187 A305188 A305189

KEYWORD

nonn,easy,mult

AUTHOR

Jianing Song, May 27 2018

STATUS

approved

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Last modified March 3 13:41 EST 2021. Contains 341762 sequences. (Running on oeis4.)