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 A000056 Order of the group SL(2,Z_n). 20
 1, 6, 24, 48, 120, 144, 336, 384, 648, 720, 1320, 1152, 2184, 2016, 2880, 3072, 4896, 3888, 6840, 5760, 8064, 7920, 12144, 9216, 15000, 13104, 17496, 16128, 24360, 17280, 29760, 24576, 31680, 29376, 40320, 31104 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The number of equivalence classes of matrices modulo n of integer matrices with determinant 1 modulo n. - Michael Somos, Mar 20 2004 24 | a(n) if n>2. - Michael Somos, Nov 15 2011 A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - Michael Somos, Jan 01 2017 REFERENCES T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 46. B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 75. LINKS T. D. Noe, Table of n, a(n) for n = 1..1000 Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003. Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)] FORMULA Multiplicative with a(p^e) = (p^2-1)*p^(3e-2). - David W. Wilson, Aug 01 2001 a(n) = A000252/phi(n), where phi is Euler totient function (cf. A000010). - Vladeta Jovovic, Oct 30 2001 a(n) = n*sum(d|n, d^2*mu(n/d)) = n*A007434(n) where A007434 is the Jordan function J_2(n). - Benoit Cloitre, May 03 2003 a(n) = A007434(n^2)/n. - Enrique Pérez Herrero, Sep 14 2010 a(n) = A007434(n^3)/n^3. - Enrique Pérez Herrero, Dec 19 2010 Dirichlet g.f. zeta(s-3)/zeta(s-1). - R. J. Mathar, Feb 27 2011 A046970(n) divides a(n). - R. J. Mathar, Mar 30 2011 EXAMPLE G.f. = x + 6*x^2 + 24*x^3 + 48*x^4 + 120*x^5 + 144*x^6 + 336*x^7 +384*x^8 + ... a(2) = 6 because [0, 1; 1, 0], [0, 1; 1, 1], [1, 0; 0, 1], [1, 0; 1, 1], [1, 1; 0, 1], [1, 1; 1, 0] are the six matrices modulo 2 with determinant 1 modulo 2. MAPLE proc(n) local b, d: b := n^3: for d from 1 to n do if irem(n, d) = 0 and isprime(d) then b := b*(1-d^(-2)): fi: od: RETURN(b): end: MATHEMATICA Table[ Fold[ If[ Mod[ n, #2 ]==0 && PrimeQ[ #2 ], #1*(1-1/#2^2), #1 ]&, n^3, Range[ n ] ], {n, 1, 35} ] Table[ n^3 Times@@(1-1/Select[ Range[ 1, n ], (Mod[ n, #1 ]==0&&PrimeQ[ #1 ])& ]^2), {n, 1, 35} ] a[ n_] := If[ n<1, 0, n Sum[ d^2 MoebiusMu[ n/d ], {d, Divisors @ n}]]; (* Michael Somos, Nov 15 2011 *) PROG (PARI) {a(n) = if( n<1, 0, n * sumdiv(n, d, d^2 * moebius(n / d)))}; /* Michael Somos, Mar 05 2008 */ CROSSREFS Row n=2 of A316623. Row sums of A316564. Cf. A000252 (number of invertible 2 X 2 matrices mod n, order of the group GL(2,Z_n)), A064767 (number of invertible 3 X 3 matrices mod n, order of the group GL(3,Z_n))), A305186 (number of invertible 4 X 4 matrices mod n, order of the group GL(4,Z_n))). Cf. A011785 (number of 3 X 3 matrices whose determinant is 1 mod n, order of the group SL(3,Z_n))), A011786 (number of 4 X 4 matrices whose determinant is 1 mod n, order of the group SL(4,Z_n))). Cf. A001766. Sequence in context: A002688 A083212 A120572 * A083170 A087081 A089973 Adjacent sequences:  A000053 A000054 A000055 * A000057 A000058 A000059 KEYWORD nonn,easy,mult AUTHOR EXTENSIONS Mathematica program from Olivier Gérard, Aug 15 1997 STATUS approved

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Last modified August 17 15:28 EDT 2018. Contains 313816 sequences. (Running on oeis4.)