A316622 -id:A316622 - cp's OEIS frontend

A002884 Number of nonsingular n X n matrices over GF(2) (order of the group GL(n,2)); order of Chevalley group A_n (2); order of projective special linear group PSL_n(2).

1, 1, 6, 168, 20160, 9999360, 20158709760, 163849992929280, 5348063769211699200, 699612310033197642547200, 366440137299948128422802227200, 768105432118265670534631586896281600, 6441762292785762141878919881400879415296000, 216123289355092695876117433338079655078664339456000

Offset: 0

N. J. A. Sloane

Also number of bases for GF(2^n) over GF(2).
Also (apparently) number of n X n matrices over GF(2) having permanent = 1. - Hugo Pfoertner, Nov 14 2003
The previous comment is true because over GF(2) permanents and determinants are the same. - Joerg Arndt, Mar 07 2008
The number of automorphisms of (Z_2)^n (the direct product of n copies of Z_2). - Peter Eastwood, Apr 06 2015
Note that n! divides a(n) since the subgroup of GL(n,2) consisting of all permutation matrices is isomorphic to S_n (the n-th symmetric group). - Jianing Song, Oct 29 2022
The number of boolean operations on n bits, or quantum operations on n qubits, that can be constructed using only CNOT (controlled NOT) gates. - David Radcliffe, Jul 06 2025

			PSL_2(2) is isomorphic to the symmetric group S_3 of order 6.

Roger W. Carter, Simple groups of Lie type. Pure and Applied Mathematics, Vol. 28, John Wiley & Sons, London-New York-Sydney, 1972. viii+331pp. MR0407163 (53 #10946). See page 2.
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.
K. J. Horadam, Hadamard matrices and their applications. Princeton University Press, Princeton, NJ, 2007. xiv+263 pp. See p. 132.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..57 (first 30 terms from T. D. Noe)
Marcus Brinkmann, Extended Affine and CCZ Equivalence up to Dimension 4, Ruhr University Bochum (2019).
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Zong Duo Dai, Solomon W. Golomb, and Guang Gong, Generating all linear orthomorphisms without repetition, Discrete Math. 205 (1999), 47-55.
P. F. Duvall, Jr. and P. W. Harley, III, A note on counting matrices, SIAM J. Appl. Math., 20 (1971), 374-377.
Nataša Ilievska and Danilo Gligoroski , Error-Detecting Code Using Linear Quasigroups, ICT Innovations 2014, Advances in Intelligent Systems and Computing Volume 311, 2015, pp 309-318.
Aaron Meyerowitz & N. J. A. Sloane, Correspondence 1979.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.
I. Strazdins, Universal affine classification of Boolean functions, Acta Applic. Math. 46 (1997), 147-167.
Index entries for sequences related to binary matrices.
Index entries for sequences related to groups.
Index to divisibility sequences.

Column k=2 of A316622 and A316623.
Cf. A000409, A000410, A002820, A005329, A006125, A028365, A046747.
Cf. A006516, A048651, A203303. Row sums of A381854.

[1] cat [(&*[2^n -2^k: k in [0..n-1]]): n in [1..15]]; // G. C. Greubel, Aug 31 2023

# First program
A002884:= n-> mul(2^n - 2^i, i=0..n-1);
seq(A002884(n), n = 0..12);
# Second program
A002884:= n-> 2^(n*(n-1)/2) * mul( 2^i - 1, i=1..n);
seq(A002884(n), n=0..12);

Table[Product[2^n-2^i,{i,0,n-1}],{n,0,13}] (* Harvey P. Dale, Aug 07 2011 *)
Table[2^(n*(n-1)/2) QPochhammer[2, 2, n] // Abs, {n, 0, 11}] (* Jean-François Alcover, Jul 15 2017 *)

a(n)=prod(i=2,n,2^i-1)<Charles R Greathouse IV, Jan 13 2012

[product(2^n -2^j for j in range(n)) for n in range(16)] # G. C. Greubel, Aug 31 2023

a(n) = Product_{i=0..n-1} (2^n-2^i).
a(n) = 2^(n*(n-1)/2) * Product_{i=1..n} (2^i - 1).
a(n) = A203303(n+1)/A203303(n). - R. J. Mathar, Jan 06 2012
a(n) = (6*a(n-1)^2*a(n-3) - 8*a(n-1)*a(n-2)^2) / (a(n-2)*a(n-3)) for n > 2. - Seiichi Manyama, Oct 20 2016
a(n) ~ A048651 * 2^(n^2). - Vaclav Kotesovec, May 19 2020
a(n) = A006125(n) * A005329(n). - John Keith, Jun 30 2021
a(n) = Product_{k=1..n} A006516(k). - Amiram Eldar, Jul 06 2025

A000252 Number of invertible 2 X 2 matrices mod n.

Original entry on oeis.org

1, 6, 48, 96, 480, 288, 2016, 1536, 3888, 2880, 13200, 4608, 26208, 12096, 23040, 24576, 78336, 23328, 123120, 46080, 96768, 79200, 267168, 73728, 300000, 157248, 314928, 193536, 682080, 138240, 892800, 393216, 633600, 470016, 967680, 373248, 1822176, 738720

Offset: 1

N. J. A. Sloane

For a prime p, a(p) = (p^2 - 1)*(p^2 - p) (this is the order of GL(2,p)). More generally a(n) is multiplicative: if the canonical factorization of n is the Product_{i=1..k} (p_i)^(e_i), then a(n) = Product_{i=1..k} (((p_i)^(2*e_i) - (p_i)^(2*e_i - 2)) * ((p_i)^(2*e_i) - (p_i)^(2*e_i - 1))). - Brian Wallace (wallacebrianedward(AT)yahoo.co.uk), Apr 05 2001, Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 18 2001
a(n) is the order of the automorphism group of the group C_n X C_n, where C_n is the cyclic group of order n. - Laszlo Toth, Dec 06 2011
Order of the group GL(2,Z_n). For n > 2, a(n) is divisible by 48. - Jianing Song, Jul 08 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.
C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, arXiv:math/0605185 [math.GR], 2006.
C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, Amer. Math. Monthly 114 (2007), no 10, 917-923.
J. Overbey, W. Traves and J. Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29 , Iss. 1, 2005.

The order of GL_2(K) for a finite field K is in sequence A059238.
Row n=2 of A316622.
Row sums of A316566.
Cf. A064767 (GL(3,Z_n)), A305186 (GL(4,Z_n)).
Cf. A000056 (SL(2,Z_n)), A011785 (SL(3,Z_n)), A011786 (SL(4,Z_n)).
Cf. A227499.

Table[n*EulerPhi[n]*Sum[d^2 MoebiusMu[n/d], {d, Divisors[n]}], {n, 21}] (* Jean-François Alcover, Apr 04 2011, after Vladeta Jovovic *)

a(n)=my(f=factor(n)[,1]); n^4*prod(i=1,#f, (1-1/f[i]^2)*(1-1/f[i])) \\ Charles R Greathouse IV, Feb 06 2017

from math import prod
from sympy import factorint
def A000252(n): return prod(p**((e<<2)-3)*(p*(p*(p-1)-1)+1) for p,e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

a(n) = n^4*Product_{primes p dividing n} (1 - 1/p^2)*(1 - 1/p) = n^4*Product_{primes p dividing n} p^(-3)*(p^2 - 1)*(p - 1). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 18 2001
Multiplicative with a(p^e) = (p - 1)^2*(p + 1)*p^(4e-3). - David W. Wilson, Aug 01 2001
a(n) = A000056(n)*phi(n), where phi is Euler totient function (cf. A000010). - Vladeta Jovovic, Oct 30 2001
Dirichlet g.f.: zeta(s - 4)*Product_{p prime} (1 - p^(1 - s)*(p^2 + p - 1)). - Álvar Ibeas, Nov 28 2017
a(n) = A227499(n) for odd n; (3/4)*A227499(n) for even n. - Jianing Song, Jul 08 2018
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = A330523 = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.5358961538283379998085... - Vaclav Kotesovec, Aug 20 2021
Sum_{n>=1} 1/a(n) = (Pi^8/3240) * Product_{p prime} (1 - 2/p^2 + 1/p^4 + 1/p^5 + 2/p^6 - 1/p^8) = 1.2059016071... . - Amiram Eldar, Dec 03 2022

More terms from David W. Wilson, Jul 21 2001

A053290 Number of nonsingular n X n matrices over GF(3).

Original entry on oeis.org

1, 2, 48, 11232, 24261120, 475566474240, 84129611558952960, 134068444202678083338240, 1923442429811445711790394572800, 248381049201184165590947520186915225600, 288678833735376059528974260112416365258106470400

Offset: 0

Stephen G Penrice, Mar 04 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..45
Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.

Column k=3 of A316622.
Cf. A002884, A003787, A027871, A047656, A053291, A053292, A053293.
Cf. A100220, A219205.

[1] cat [&*[(3^n - 3^k): k in [0..n-1]]: n in [1..9]]; // Bruno Berselli, Jan 28 2013

Table[Product[3^n - 3^k, {k, 0, n - 1}], {n, 0, 10}] (* Geoffrey Critzer, Jan 26 2013; edited by Vincenzo Librandi, Jan 28 2013 *)

for(n=0,10, print1(prod(k=0,n-1, 3^n - 3^k), ", ")) \\ G. C. Greubel, May 31 2018

a(n) = Product_{k=0..n-1}(3^n-3^k). - corrected by Michel Marcus, Sep 18 2015
a(n) = A047656(n)*A027871(n). - Bruno Berselli, Jan 30 2013
From Amiram Eldar, Jul 06 2025: (Start)
a(n) = Product_{k=1..n} A219205(k).
a(n) ~ c * 3^(n^2), where c = A100220. (End)

More terms from Vladeta Jovovic, Mar 16 2000

A053292 Number of nonsingular n X n matrices over GF(5).

Original entry on oeis.org

1, 4, 480, 1488000, 116064000000, 226614960000000000, 11064475422000000000000000, 13506266841692625000000000000000000, 412177498341354683437500000000000000000000000

Offset: 0

Stephen G Penrice, Mar 04 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..35
Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.

Column k=5 of A316622.

Cf. A002884, A003789, A027872, A053290, A053291, A053293, A109345, A100222.

[1] cat [&*[(5^n - 5^k): k in [0..n-1]]: n in [1..8]]; // Bruno Berselli, Jan 28 2013

Table[Product[5^n - 5^k, {k,0,n-1}], {n,0,10}] (* Geoffrey Critzer, Jan 26 2013 *)

for(n=0,10, print1(prod(k=0,n-1, 5^n - 5^k), ", ")) \\ G. C. Greubel, May 31 2018

a(n) = (5^n - 1)*(5^n - 5)*...*(5^n - 5^(n-1)).
a(n) = A109345(n)*A027872(n). - Bruno Berselli, Jan 30 2013
a(n) ~ c * 5^(n^2), where c = A100222. - Amiram Eldar, Jul 06 2025

More terms from Vladeta Jovovic, Mar 16 2000

A053293 Number of nonsingular n X n matrices over GF(7).

Original entry on oeis.org

1, 6, 2016, 33784128, 27811094169600, 1122211189922928537600, 2218959336124989671614429593600, 214992513152176999576908105619651923148800, 1020690003311610463765638355505358381593396977336320000, 237443634207909205360438080389756681126654524500073656592021585920000

Offset: 0

Stephen G Penrice, Mar 04 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..30
Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.

Column k=7 of A316622.

Cf. A002884, A003790, A027875, A053290, A053291, A053292, A109493, A132035.

[1] cat [&*[(7^n - 7^k): k in [0..n-1]]: n in [1..7]]; // Bruno Berselli, Jan 28 2013

Table[Product[7^n - 7^k, {k, 0, n-1}], {n, 0, 10}] (* Vincenzo Librandi, Jan 28 2013 *)

for(n=0,10, print1(prod(k=0,n-1, 7^n - 7^k), ", ")) \\ G. C. Greubel, May 31 2018

a(n) = (7^n - 1)*(7^n - 7)*...*(7^n - 7^(n-1)).
a(n) = A109493(n)*A027875(n). - Bruno Berselli, Jan 30 2013
a(n) ~ c * 7^(n^2), where c = A132035. - Amiram Eldar, Jul 06 2025

More terms from Vladeta Jovovic, Mar 16 2000

A064767 Order of automorphism group of the group C_n X C_n X C_n (where C_n is the cyclic group of order n).

Original entry on oeis.org

1, 168, 11232, 86016, 1488000, 1886976, 33784128, 44040192, 221079456, 249984000, 2124276000, 966131712, 9726417792, 5675733504, 16713216000, 22548578304, 111203278848, 37141348608, 304812862560, 127991808000

Offset: 1

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Oct 24 2001

Also number of 3 X 3 invertible matrices over the ring Z/nZ. - Max Alekseyev, Nov 02 2007

Order of the group GL(3,Z_n). For n > 2, a(n) is divisible by 96. - 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.
C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, arXiv:math/0605185 [math.GR], 2006.
C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, Amer. Math. Monthly 114 (2007), no 10, 917-923.
J. Overbey, W. Traves and J. Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29 , Iss. 1, 2005.
Index entries for sequences related to groups.

Row n=3 of A316622.
Cf. A000010, A064803.
Cf. A000252 (GL(2,Z_n)), A305186 (GL(4,Z_n)).
Cf. A000056 (SL(2,Z_n)), A011785 (SL(3,Z_n)), A011786 (SL(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]]; a[1] = 1; Array[a, 20] (* 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))); \\ Michel Marcus, Jun 30 2015

a(n,f=factor(n))=prod(i=1,#f~, ((1 - 1/f[i,1]^3)*(1 - 1/f[i,1]^2)*(1 - 1/f[i,1])))*n^9 \\ Charles R Greathouse IV, Mar 04 2025

from math import prod
from sympy import factorint
def A064767(n): return prod(p**(3*(3*e-2))*(p*(p*(p**2*(p*(p-1)-1)+1)+1)-1) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

a(n) = phi(n)*A011785(n). - Vladeta Jovovic, Oct 29 2001
a(n) = n^9*Product_{primes p dividing n} ((1 - 1/p^3)*(1 - 1/p^2)*(1 - 1/p)). This also gives a formula for A011785.
Multiplicative with a(p^e) = p^(9*e-6)*(p^3 - 1)*(p^2 - 1)*(p - 1). - Vladeta Jovovic, Nov 18 2001
Sum_{k=1..n} a(k) ~ c * n^10, where c = (1/10) * Product_{p prime} ((p^7 - p^5 - p^4 + p^2 + p - 1)/p^7) =  0.05123382571... . - Amiram Eldar, Oct 23 2022

More terms from Vladeta Jovovic, Nov 18 2001

A305186 Number of invertible 4 X 4 matrices mod n.

Original entry on oeis.org

1, 20160, 24261120, 1321205760, 116064000000, 489104179200, 27811094169600, 86586540687360, 1044361663787520, 2339850240000000, 41393302251840000, 32053931488051200, 610296923230525440, 560671658459136000, 2815842631680000000, 5674535530486824960

Offset: 1

Jianing Song, May 27 2018

Order of the group GL(4,Z_n).
Order of the automorphism group of the group (C_n)^4, where C_n is the cyclic group of order n.
For n > 2, a(n) is divisible by 23040.

Jianing Song, Table of n, a(n) for n = 1..10000
C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, arXiv:math/0605185 [math.GR], 2006.
C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, Amer. Math. Monthly 114 (2007), no 10, 917-923.
J. Overbey, W. Traves and J. Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1, 2005.

Row n=4 of A316622.
Cf. A000252 (GL(2,Z_n)), A064767 (GL(3,Z_n)).
Cf. A000056 (SL(2,Z_n)), A011785 (SL(3,Z_n)), A011786 (SL(4,Z_n)).
Cf. A000010.

{1}~Join~Array[#^16*Product[(1 - 1/p^4) (1 - 1/p^3) (1 - 1/p^2) (1 - 1/p), {p, FactorInteger[#][[All, 1]]}] &, 12, 2] (* Michael De Vlieger, May 27 2018 *)

a(n)=my(f=factor(n)[, 1]); n^16*prod(i=1, #f, (1-1/f[i]^4)*(1-1/f[i]^3)*(1-1/f[i]^2)*(1-1/f[i]))

from math import prod
from sympy import factorint
def A305186(n): return prod(p**((e<<3)-5<<1)*(p*(p*(p**3*(p**3*(p*(p-1)-1)+2)-1)-1)+1) for p,e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

Multiplicative with a(p^e) = (p - 1)*(p^2 - 1)*(p^3 - 1)*(p^4 - 1)*p^(16*e-10).
a(n) = n^16*Product_{primes p dividing n} (1 - 1/p^4)*(1 - 1/p^3)*(1 - 1/p^2)*(1 - 1/p).
a(n) = phi(n)*A011786(n) = A000010(n)*A011786(n).
Sum_{k=1..n} a(k) ~ c * n^17, where c = (1/17) * Product_{p prime} ((p^11 - p^9 - p^8 + 2*p^5 - p^2 - p + 1)/p^11) = 0.02958150406... . - Amiram Eldar, Oct 23 2022

A065128 Number of invertible n X n matrices mod 4 (i.e., over the ring Z_4).

Original entry on oeis.org

1, 2, 96, 86016, 1321205760, 335522845163520, 1385295986380096143360, 92239345887620476544860815360, 98654363640526679389774053813465907200, 1691558770638735027870457216848672340872423014400, 464518059995994038184379206447729320401459864818351813427200

Offset: 0

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 14 2001

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.

Column k=4 of A316622.

Cf. A048651, A053291, A060195.

a[n_] := 4^(n^2)*Product[1 - 1/2^k, {k, 1, n} ]; Table[ a[n], {n, 0, 10} ]

for(n=1,11,print(4^(n^2)*prod(k=1,n,(1-1/2^k))))

a(n) = 4^(n^2) * Product_{k=1..n} (1 - 1/2^k).
a(n) = 2^(n^2) * A002884(n). - Geoffrey Critzer, Feb 04 2018
From Amiram Eldar, Jul 06 2025: (Start)
a(n) = Product_{k=1..n} 2*A060195(k).
a(n) ~ c * 4^(n^2), where c = A048651. (End)

More terms from Robert G. Wilson v and Jason Earls, Nov 16 2001

A316623 Array read by antidiagonals: T(n,k) is the order of the group SL(n,Z_k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 24, 168, 1, 1, 1, 48, 5616, 20160, 1, 1, 1, 120, 43008, 12130560, 9999360, 1, 1, 1, 144, 372000, 660602880, 237783237120, 20158709760, 1, 1, 1, 336, 943488, 29016000000, 167761422581760, 42064805779476480, 163849992929280, 1

Offset: 0

Andrew Howroyd, Jul 08 2018

All rows are multiplicative.
Equivalently, the number of n X n matrices mod k with determinant 1.
Also, for k prime (but not higher prime powers) the number of n X n matrices over GF(k) with determinant 1.

			Array begins:
==============================================================
n\k| 1       2        3         4           5           6
---+----------------------------------------------------------
0  | 1       1        1         1           1            1 ...
1  | 1       1        1         1           1            1 ...
2  | 1       6       24        48         120          144 ...
3  | 1     168     5616     43008      372000       943488 ...
4  | 1   20160 12130560 660602880 29016000000 244552089600 ...
5  | 1 9999360 ...
...

R. P. Brent and B. D. McKay, Determinants and ranks of random matrices over Zm, Discrete Mathematics 66 (1987) pp. 35-49.
J. M. Lockhart and W. P. Wardlaw, Determinants of Matrices over the Integers Modulo m, Mathematics Magazine, Vol. 80, No. 3 (Jun., 2007), pp. 207-214.
The Group Properties Wiki, Order formulas for linear groups

Rows n=2..4 are A000056, A011785, A011786.
Columns k=2..5, 7 are A002884, A003787, A011787, A003789, A003790.
Cf. A316622.

T:=function(n,k) if k=1 or n=0 then return 1; else return Order(SL(n, Integers mod k)); fi; end;
for n in [0..5] do Print(List([1..6], k->T(n,k)), "\n"); od;

T[n_, k_] := If[k == 1 || n == 0, 1, k^(n^2-1) Product[1 - p^-j, {p, FactorInteger[k][[All, 1]]}, {j, 2, n}]];
Table[T[n-k+1, k], {n, 0, 8}, {k, n+1, 1, -1}] // Flatten (* Jean-François Alcover, Sep 19 2019 *)

T(n,k)={my(f=factor(k)); if(n<1, n==0, k^(n^2-1) * prod(i=1, #f~, my(p=f[i,1]); prod(j=2, n, (1 - p^(-j)))))}

T(n,p^e) = (p^e)^(n^2-1) * Product_{j=2..n} (1 - 1/p^j) for prime p, n > 0.

A065498 Number of invertible n X n matrices mod 6 (i.e., over the ring Z_6).

Original entry on oeis.org

1, 2, 288, 1886976, 489104179200, 4755360379856486400, 1695944421638473850132889600, 21967113634648374162210646578639667200, 10286692771039109536373764545035369981946101760000, 173770439600109774111384717714984362383506603790098046648320000

Offset: 0

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 25 2001

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.

Column k=6 of A316622.

Cf. A002884, A048651, A053290, A065128, A100220.

a[n_] := 6^(n^2)*Product[(1 - 1/2^k)*(1 - 1/3^k), { k, 1, n} ]; Table[ a[n], {n, 0, 9} ]

a(n) = 6^(n^2) * Product_{k=1..n} ((1 - 1/2^k)(1 - 1/3^k)).
a(n) = A002884(n)*A053290(n). - Geoffrey Critzer, Jan 26 2018
a(n) ~ c * 6^(n^2), where c = A048651 * A100220 = 0.161757743053... . - Amiram Eldar, Jul 06 2025

More terms from Robert G. Wilson v, Nov 28 2001

cp's OEIS Frontend

A002884 Number of nonsingular n X n matrices over GF(2) (order of the group GL(n,2)); order of Chevalley group A_n (2); order of projective special linear group PSL_n(2).

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A000252 Number of invertible 2 X 2 matrices mod n.

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A053290 Number of nonsingular n X n matrices over GF(3).

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A053292 Number of nonsingular n X n matrices over GF(5).

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A053293 Number of nonsingular n X n matrices over GF(7).

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A064767 Order of automorphism group of the group C_n X C_n X C_n (where C_n is the cyclic group of order n).

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