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A011785 Number of 3 X 3 matrices whose determinant is 1 mod n. 8
1, 168, 5616, 43008, 372000, 943488, 5630688, 11010048, 36846576, 62496000, 212427600, 241532928, 810534816, 945955584, 2089152000, 2818572288, 6950204928, 6190224768, 16934047920, 15998976000, 31621943808, 35687836800 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Order of the group SL(3,Z_n). For n > 2, a(n) is divisible by 48. - Jianing Song, Nov 24 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.

FORMULA

Multiplicative with a(p^e) = p^(8*e-5)*(p^3 - 1)*(p^2 - 1). - Vladeta Jovovic, Nov 18 2001

For a formula see A064767.

a(n) = A046970(n)*A063453(n)*A000578(n)*A003557(n)^5. - R. J. Mathar, Mar 30 2011

a(n) = A064767(n)/phi(n). - Jianing Song, Nov 24 2018

Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^5/((p-1)^3 * (p+1)^2 * (p^2 + p + 1) * (p^6 + p^4 + p^2 + 1))) = 1.0061577672748872278355775942508642214184417621389767880397578015151659965... - Vaclav Kotesovec, Sep 19 2020

MATHEMATICA

a[n_] := (n^9*Times @@ Function[p, (1 - 1/p^3)*(1 - 1/p^2)*(1 - 1/p)] /@ FactorInteger[n][[All, 1]])/EulerPhi[n]; a[1] = 1; Array[a, 30] (* Jean-Fran├žois Alcover, Mar 21 2017 *)

PROG

(PARI) a(n) = n^9*prod(k=2, n, if (!isprime(k) || (n % k), 1, (1-1/k^3)*(1-1/k^2)*(1-1/k)))/eulerphi(n); \\ Michel Marcus, Jun 30 2015

CROSSREFS

Cf. A003800.

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

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

Sequence in context: A223243 A282614 A003807 * A227433 A003800 A278010

Adjacent sequences:  A011782 A011783 A011784 * A011786 A011787 A011788

KEYWORD

nonn,mult

AUTHOR

Benjamin T. Love (benlove(AT)preston.polaristel.net)

EXTENSIONS

More terms from John W. Layman, Feb 16 2001

Further terms from Vladeta Jovovic, Oct 29 2001

STATUS

approved

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Last modified March 3 12:59 EST 2021. Contains 341762 sequences. (Running on oeis4.)