A011785 Number of 3 X 3 matrices whose determinant is 1 mod n.

1, 168, 5616, 43008, 372000, 943488, 5630688, 11010048, 36846576, 62496000, 212427600, 241532928, 810534816, 945955584, 2089152000, 2818572288, 6950204928, 6190224768, 16934047920, 15998976000, 31621943808, 35687836800

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

Sum_{k=1..n} a(k) ~ c * n^9, where c = (1/9) * Product_{p prime} (1 - (p^3 + p^2 -1)/p^6) = 0.08630488937... . - Amiram Eldar, Oct 23 2022

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-François 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. A000010, A000578, A003557, A003800, A046970, A063453.

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 A373501

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