A003787 -id:A003787 - cp's OEIS frontend

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

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

A003793 Order of (usually) simple Chevalley group A_n (3).

Original entry on oeis.org

1, 12, 5616, 6065280, 237783237120, 21032402889738240, 67034222101339041669120, 480860607452861427947598643200, 124190524600592082795473760093457612800

Offset: 0

N. J. A. Sloane

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.

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Robert Steinberg, Lectures on Chevalley Groups, Dept. of Mathematics, Yale University, 1967, p. 130-131.
Index entries for sequences related to groups.

Cf. A003787, A003794, A003795, A003797, A003798, A003799, A100220.

f[m_, n_] := Block[{g, x, y}, g[x_, y_] := x^(y (y + 1)/2) Product[x^k - 1, {k, 2, y + 1}]; g[m, n]/GCD[n + 1, m - 1]]; f[3, #] & /@ Range[0, 8] (* Michael De Vlieger, Sep 18 2015 *)

a(n) = a(3,n) where a(q,n) = A(q,n) / gcd(n+1, q-1) and A(q,n) is defined in A003787. - Sean A. Irvine, Sep 18 2015

a(n) ~ c * gcd(n, 2) * 3^(n^2 + 2*n), where c = (3/4) * A100220. - Amiram Eldar, Jul 11 2025

A003789 Order of universal Chevalley group A_n (5).

Original entry on oeis.org

1, 120, 372000, 29016000000, 56653740000000000, 2766118855500000000000000, 3376566710423156250000000000000000, 103044374585338670859375000000000000000000000

Offset: 0

N. J. A. Sloane

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.

Index entries for sequences related to groups.

Cf. A003787, A003788, A003790, A003791, A003792, A053292, A100222.

[&*[(5^n-5^k): k in [0..n-1]]/4: n in [1..8]]; // Vincenzo Librandi, Sep 19 2015

f[m_, n_] := m^(n (n + 1)/2) Product[m^k - 1, {k, 2, n + 1}]; f[5, #] & /@ Range[0, 7] (* Michael De Vlieger, Sep 18 2015 *)

Numbers so far appear to equal A053292(n)/4. - Ralf Stephan, Mar 30 2004
a(n) = A(5,n) where A(q,n) is defined in A003787. - Sean A. Irvine, Sep 18 2015
a(n) ~ c * 5^(n*(n+2)), where c = (5/4) * A100222 = 0.950415994839... . - Amiram Eldar, Jul 07 2025

A003790 Order of universal Chevalley group A_n (7).

Original entry on oeis.org

1, 336, 5630688, 4635182361600, 187035198320488089600, 369826556020831611935738265600, 35832085525362833262818017603275320524800, 170115000551935077294273059250893063598899496222720000

Offset: 0

N. J. A. Sloane

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.

Index entries for sequences related to groups.

Cf. A003787, A003788, A003789, A003791, A003792,A053293, A100220.

[&*[(7^n - 7^k): k in [0..n-1]]/6: n in [1..10]]; // Vincenzo Librandi, Sep 19 2015

f[m_, n_] := m^(n (n + 1)/2) Product[m^k - 1, {k, 2, n + 1}]; f[7, #] & /@ Range[0, 7] (* Michael De Vlieger, Sep 18 2015 *)

Numbers so far appear to equal A053293(n)/6. - Ralf Stephan, Mar 30 2004
a(n) = A(7,n) where A(q,n) is defined in A003787. - Sean A. Irvine, Sep 18 2015
a(n) ~ c * 7^(n*(n+2)), where c = (7/6) * A100220 = 0.840189116891... . - Amiram Eldar, Jul 07 2025

a(7) from Sean A. Irvine, Sep 18 2015

A003792 Order of universal Chevalley group A_n (9).

Original entry on oeis.org

1, 720, 42456960, 203039372390400, 78660280796419613491200, 2468438315722201136962330755072000, 6274437692242927471137606015213542491815936000, 1291851049702792234730057308758464452124128263449062932480000

Offset: 0

N. J. A. Sloane

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.

Index entries for sequences related to groups.

Cf. A003787, A003788, A003789, A003790, A003791, A052497, A132037.

[&*[(9^n - 9^k): k in [0..n-1]]/8: n in [1..10]]; // Vincenzo Librandi, Sep 19 2015

f[m_, n_] := m^(n (n + 1)/2) Product[m^k - 1, {k, 2, n + 1}];
f[9, #] & /@ Range[0, 7] (* Michael De Vlieger, Sep 18 2015 *)

Numbers so far appear to equal A052497(n)/8. - Ralf Stephan, Mar 30 2004
a(n) = A(9,n) where A(q,n) is defined in A003787. - Sean A. Irvine, Sep 18 2015
a(n) ~ c * 9^(n*(n+2)), where c = (9/8) * A132037 = 0.9861303982904... . - Amiram Eldar, Jul 07 2025

a(7) from Sean A. Irvine, Sep 18 2015

A003788 Order of universal Chevalley group A_n (4).

Original entry on oeis.org

1, 60, 60480, 987033600, 258492255436800, 1083930404878024704000, 72736898347485916060188672000, 78099458182389588115529148326215680000, 1341733356588640095264385107865053233298800640000

Offset: 0

N. J. A. Sloane

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.

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Index entries for sequences related to groups.

Cf. A003787, A003789, A003790, A003791, A003792, A053291, A100221.

[&*[(4^n - 4^k): k in [0..n-1]]/3: n in [1..8]]; // Vincenzo Librandi, Sep 19 2015

f[m_, n_] := m^(n (n + 1)/2) Product[m^k - 1, {k, 2, n + 1}];
f[4, #] & /@ Range[0, 8] (* Michael De Vlieger, Sep 18 2015 *)

Numbers so far appear to equal A053291(n)/3. - Ralf Stephan, Mar 30 2004
a(n) = A(4,n) where A(q,n) is defined in A003787. - Sean A. Irvine, Sep 18 2015
a(n) ~ c * 4^(n*(n+2)), where c = (4/3) * A100221 = 0.918050049493... . - Amiram Eldar, Jul 07 2025

One more term from Sean A. Irvine, Sep 18 2015

A003791 Order of universal Chevalley group A_n (8).

Original entry on oeis.org

1, 504, 16482816, 34558531338240, 4638226007491010887680, 39841906041871272087686291128320, 21903309038581548352789123727634573903790080

Offset: 0

N. J. A. Sloane

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.

Index entries for sequences related to groups.

Cf. A003787, A003788, A003789, A003790, A003792, A052496, A132036.

[&*[(8^n - 8^k): k in [0..n-1]]/7: n in [1..8]]; // Vincenzo Librandi, Sep 19 2015

f[m_, n_] := m^(n (n + 1)/2) Product[m^k - 1, {k, 2, n + 1}]; f[8, #] & /@ Range[0, 6] (* Michael De Vlieger, Sep 18 2015 *)

Numbers so far appear to equal A052496(n)/7. - Ralf Stephan, Mar 30 2004
a(n) = A(8,n) where A(q,n) is defined in A003787. - Sean A. Irvine, Sep 18 2015
a(n) ~ c * 8^(n*(n+2)), where c = (8/7) * A132036 = 0.982178279315... . - Amiram Eldar, Jul 07 2025

A003801 Order of universal Chevalley group A_3 (q) (or D_3 (q)), q = prime power.

Original entry on oeis.org

20160, 12130560, 987033600, 29016000000, 4635182361600, 34558531338240, 203039372390400, 4139330225184000, 50858076935877120, 1148120010326016000, 2851903720876769280, 15136750711925049600

Offset: 1

N. J. A. Sloane

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.

f[m_, n_] := m^(n (n + 1)/2) Product[m^k - 1, {k, 2, n + 1}]; f[#, 3] & /@ Select[Range[2, 22], PrimePowerQ] (* Michael De Vlieger, Sep 18 2015 *)

a(n) = A(A000961(n+1),3) where A(q,n) is defined in A003787. - Sean A. Irvine, Sep 18 2015

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.

A003800 Order of universal Chevalley group A_2 (q), q = prime power.

Original entry on oeis.org

168, 5616, 60480, 372000, 5630688, 16482816, 42456960, 212427600, 810534816, 4277145600, 6950204928, 16934047920, 78156525216, 152334000000, 282027786768, 499631102880, 851974934400, 1098404364288

Offset: 1

N. J. A. Sloane

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.

f[m_, n_] := m^(n (n + 1)/2) Product[m^k - 1, {k, 2, n + 1}]; f[#, 2] & /@ Select[Range[2, 36], PrimePowerQ] (* Michael De Vlieger, Sep 18 2015 *)

a(n) = A(A000961(n+1),2) where A(q,n) is defined in A003787. - Sean A. Irvine, Sep 18 2015

cp's OEIS Frontend

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

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A003793 Order of (usually) simple Chevalley group A_n (3).

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A003789 Order of universal Chevalley group A_n (5).

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A003790 Order of universal Chevalley group A_n (7).

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A003792 Order of universal Chevalley group A_n (9).

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A003788 Order of universal Chevalley group A_n (4).

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A003791 Order of universal Chevalley group A_n (8).

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