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A000056 Order of the group SL(2,Z_n). 24

%I

%S 1,6,24,48,120,144,336,384,648,720,1320,1152,2184,2016,2880,3072,4896,

%T 3888,6840,5760,8064,7920,12144,9216,15000,13104,17496,16128,24360,

%U 17280,29760,24576,31680,29376,40320,31104,50616,41040,52416,46080,68880,48384,79464

%N Order of the group SL(2,Z_n).

%C The number of equivalence classes of matrices modulo n of integer matrices with determinant 1 modulo n. - _Michael Somos_, Mar 20 2004

%C 24 | a(n) if n > 2. - _Michael Somos_, Nov 15 2011

%C 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

%C The group SL(2,Z_2) is isomorphic to the symmetric group S_3. - _Bernard Schott_, Mar 15 2020

%D T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 46.

%D B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 75.

%H T. D. Noe, <a href="/A000056/b000056.txt">Table of n, a(n) for n = 1..1000</a>

%H Ed Pegg, Jr., <a href="http://www.mathpuzzle.com/MAA/07-Sequence%20Pictures/mathgames_12_08_03.html">Sequence Pictures</a>, Math Games column, Dec 08 2003.

%H Ed Pegg, Jr., <a href="/A000043/a000043_2.pdf">Sequence Pictures</a>, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>

%F Multiplicative with a(p^e) = (p^2 - 1)*p^(3e-2). - _David W. Wilson_, Aug 01 2001

%F a(n) = A000252(n)/phi(n), where phi is Euler totient function (cf. A000010). - _Vladeta Jovovic_, Oct 30 2001

%F 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

%F a(n) = A007434(n^2)/n. - _Enrique Pérez Herrero_, Sep 14 2010

%F a(n) = A007434(n^3)/n^3. - _Enrique Pérez Herrero_, Dec 19 2010

%F Dirichlet g.f. zeta(s-3)/zeta(s-1). - _R. J. Mathar_, Feb 27 2011

%F A046970(n) divides a(n). - _R. J. Mathar_, Mar 30 2011

%F Sum_{k=1..n} a(k) ~ n^4 / (4*Zeta(3)). - _Vaclav Kotesovec_, Jan 30 2019

%F Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^2 / ((p-1)^2 * (p+1) * (p^2 + p + 1))) = 1.258448350408311046314826069717731136828991478925039589864338603650639811... - _Vaclav Kotesovec_, Sep 19 2020

%e G.f. = x + 6*x^2 + 24*x^3 + 48*x^4 + 120*x^5 + 144*x^6 + 336*x^7 +384*x^8 + ...

%e 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.

%p 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:

%t (* From _Olivier Gérard_, Aug 15 1997: (Start) *)

%t Table[ Fold[ If[ Mod[ n, #2 ]==0 && PrimeQ[ #2 ], #1*(1-1/#2^2), #1 ]&, n^3, Range[ n ] ], {n, 1, 35} ]

%t Table[ n^3 Times@@(1-1/Select[ Range[ 1, n ], (Mod[ n, #1 ]==0&&PrimeQ[ #1 ])& ]^2), {n, 1, 35} ] (* End *)

%t a[ n_] := If[ n<1, 0, n Sum[ d^2 MoebiusMu[ n/d ], {d, Divisors @ n}]]; (* _Michael Somos_, Nov 15 2011 *)

%t Table[ n DirichletConvolve[ MoebiusMu[m], m^2, m, n], {n, 1, 35}] (* _Li Han_, Mar 15 2020 *)

%t a[n_] := #.RotateLeft[#] & @ Sort[Mod[ Outer[Times, Range[n], Range[n]], n] // Flatten // Tally][[;; , 2]]

%t Table[a[n], {n, 1, 35}] (* Li Han, Mar 15 2020 *)

%o (PARI) {a(n) = if( n<1, 0, n * sumdiv(n, d, d^2 * moebius(n / d)))}; /* _Michael Somos_, Mar 05 2008 */

%Y Cf. A001766.

%Y Row n=2 of A316623.

%Y Row sums of A316564.

%Y Cf. A000252 (GL(2,Z_n)), A064767 (GL(3,Z_n)), A305186 (GL(4,Z_n)).

%Y Cf. A011785 (SL(3,Z_n)), A011786 (SL(4,Z_n)).

%K nonn,easy,mult

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Vaclav Kotesovec_, Sep 19 2020

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Last modified March 3 13:15 EST 2021. Contains 341762 sequences. (Running on oeis4.)