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 A000252 Number of invertible 2 X 2 matrices mod n. 33
 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_{i=1..k} (p_i)^(e_i), then a(n) = Product_{i=1..k} (((p_i)^(2*e_i) - (p_i)^(2*e_i - 2)) * ((p_i)^(2*e_i) - (p_i)^(2*e_i - 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 Order of the group GL(2,Z_n). For n > 2, a(n) is divisible by 48. - Jianing Song, Jul 08 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. FORMULA a(n) = n^4*Product_{primes p dividing n} (1 - 1/p^2)*(1 - 1/p) = n^4*Product_{primes p dividing n} p^(-3)*(p^2 - 1)*(p - 1). - 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 Dirichlet g.f.: zeta(s - 4)*Product_{p prime} (1 - p^(1 - s)*(p^2 + p - 1)). - Álvar Ibeas, Nov 28 2017 a(n) = A227499(n) for odd n; (3/4)*A227499(n) for even n. - Jianing Song, Jul 08 2018 Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = A330523 = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.5358961538283379998085... - Vaclav Kotesovec, Aug 20 2021 Sum_{n>=1} 1/a(n) = (Pi^8/3240) * Product_{p prime} (1 - 2/p^2 + 1/p^4 + 1/p^5 + 2/p^6 - 1/p^8) = 1.2059016071... . - Amiram Eldar, Dec 03 2022 MATHEMATICA Table[n*EulerPhi[n]*Sum[d^2 MoebiusMu[n/d], {d, Divisors[n]}], {n, 21}] (* Jean-François 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 order of GL_2(K) for a finite field K is in sequence A059238. Row n=2 of A316622. Row sums of A316566. Cf. A064767 (GL(3,Z_n)), A305186 (GL(4,Z_n)). Cf. A000056 (SL(2,Z_n)), A011785 (SL(3,Z_n)), A011786 (SL(4,Z_n)). Cf. A227499. Sequence in context: A354067 A357430 A323138 * A078237 A274131 A341683 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 December 9 07:37 EST 2023. Contains 367689 sequences. (Running on oeis4.)