A003800 -id:A003800 - cp's OEIS frontend

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

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

Order of the group SL(3,Z_n). For n > 2, a(n) is divisible by 48. - Jianing Song, Nov 24 2018

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.

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)).

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 *)

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

from math import prod
from sympy import factorint
def A011785(n): return prod(p**((e<<3)-5)*(p**2*(p*(p-1)*(p+1)-1)+1) for p,e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

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

More terms from John W. Layman, Feb 16 2001

Further terms from Vladeta Jovovic, Oct 29 2001

A373501 Size of the collineation group of classical projective planes of prime power order q.

Original entry on oeis.org

168, 5616, 120960, 372000, 5630688, 49448448, 84913920, 212427600, 810534816, 17108582400, 6950204928, 16934047920, 78156525216, 304668000000, 846083360304, 499631102880, 851974934400, 5492021821440, 3509844434208, 7980059337600, 11681731985616, 23800278205248

Offset: 1

Ralf Goertz, Jun 07 2024

nonn

a(A246655(n)) is the size of the collineation group of the classical projective plane of order q=p^k. It is also known as the projective semilinear group, PGammaL(3,q), the semidirect product of PGL(3,q) (whose order is probably given by A003800) with the group of field automorphisms of F(q). The latter is the cyclic group of order k. Therefore, |PGammaL(3,p^k)|=|PGL(3,p^k)|*k.

			Take for example the first value 168 which refers to the number of automorphisms of the Fano plane (q=2). Its v=7 (=q^2+q+1) lines are subsets of size 3 (=q+1) of a set of v points. Using 0,1,...,6 to label these points, one way of enumerating the lines is depicted in the first column of the following table:
            (0 1 2 3 4 5 6)   (0 6)(3 5)
  {0,1,3}       {1,2,4}        {6,1,5}
  {1,2,4}       {2,3,5}        {1,2,4}
  {2,3,5}       {3,4,6}        {2,5,3}
  {3,4,6}       {4,5,0}        {5,4,0}
  {4,5,0}       {5,6,1}        {4,3,6}
  {5,6,1}       {6,0,2}        {3,0,1}
  {6,0,2}       {0,1,3}        {0,6,2}
Note that any two distinct lines have exactly 1 point in common. Applying one of the 7!=5040 possible permutations of the points obviously doesn't change that fact. However, exactly 168 of these permutations lead to the same set of subsets. One such permutation is the full cycle (0,1,2,3,4,5,6) whose action can bee seen in the second column. It also permutes the lines cyclically by mapping line i to line i+1 (mod v). Another one is the cycle product (0 6)(3 5) in the third column. It swaps lines 1 and 6 and lines 4 and 5 and leaves the other three lines fixed.

A. Beutelspacher and U. Rosenbaum, Projective Geometry: From Foundations to Applications, Cambridge University Press, 1998, pages 118-132.
D. R. Hughes and F. C. Piper, Projective Planes, Springer, 1973.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Projective linear group, order of PGL(3,q) and PGammaL(3,q).

Cf. A373502 for the size of a complete set of classical projective planes using a given set of q^2+q+1 points.

Cf. A335866 for the number of projective planes whose lines are cyclic difference sets.

Map[PrimeOmega[#]*#^3*(#^2+#+1)*(#^2-1)*(#-1) &, Select[Range[50], PrimePowerQ]] (* Paolo Xausa, Aug 01 2024 *)

a=(q)->bigomega(q)*(q^3-1)*(q^3-q)*(q^3-q^2)/(q-1) \\ q=A246655(n)

a(n) = Omega(q)*(q^3-1)*(q^3-q)*(q^3-q^2)/(q-1) where q = A246655(n).

Data corrected by Paolo Xausa, Aug 02 2024

cp's OEIS Frontend

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

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