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A000252 Number of invertible 2 X 2 matrices mod n. 22
1, 6, 48, 96, 480, 288, 2016, 1536, 3888, 2880, 13200, 4608, 26208, 12096, 23040, 24576, 78336, 23328, 123120, 46080, 96768, 79200, 267168, 73728, 300000, 157248, 314928, 193536, 682080, 138240, 892800, 393216, 633600, 470016, 967680, 373248 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

For a prime p, a(p) = (p^2 - 1)*(p^2 - p) (this is the order of GL(2,p)). More generally a(n) is multiplicative: if the canonical factorization of n is the product of p^e(p) over primes p then a(n) = product ((p^(2*e(p)) - p^(2*e(p) - 2)) * (p^(2*e(p)) - p^(2*e(p) - 1))). - Brian Wallace (wallacebrianedward(AT)yahoo.co.uk), Apr 05 2001, Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 18 2001

a(n) is the order of the automorphism group of the group C_n X C_n, where C_n is the cyclic group of order n. - Laszlo Toth, Dec 06 2011

LINKS

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

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

a(n) = n^4 * product (1-1/p^2)*(1-1/p) = n^4 * product p^(-3)(p^2 - 1)*(p - 1) where the product is over all the primes p that divide n. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 18 2001

Multiplicative with a(p^e) = (p-1)^2*(p+1)*p^(4e-3). - David W. Wilson, Aug 01 2001

a(n) = A000056(n)*phi(n), where phi is Euler totient function (cf. A000010). - Vladeta Jovovic, Oct 30 2001

MATHEMATICA

Table[n*EulerPhi[n]*Sum[d^2 MoebiusMu[n/d], {d, Divisors[n]}], {n, 21}] (* Jean-Fran├žois Alcover, Apr 04 2011, after Vladeta Jovovic *)

PROG

(PARI) a(n)=my(f=factor(n)[, 1]); n^4*prod(i=1, #f, (1-1/f[i]^2)*(1-1/f[i])) \\ Charles R Greathouse IV, Feb 06 2017

CROSSREFS

The number of 2 X 2 matrices mod n with determinant 1 is A000056. The order of GL_2(K) for a finite field K is in sequence A059238.

Cf. A011785, A064767.

Sequence in context: A104256 A289211 A192887 * A078237 A274131 A259121

Adjacent sequences:  A000249 A000250 A000251 * A000253 A000254 A000255

KEYWORD

nonn,easy,nice,mult

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from David W. Wilson, Jul 21 2001

STATUS

approved

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Last modified November 18 02:54 EST 2017. Contains 294840 sequences.