A065128 -id:A065128 - cp's OEIS frontend

A316622 Array read by antidiagonals: T(n,k) is the order of the group GL(n,Z_k).

1, 1, 1, 1, 1, 1, 1, 2, 6, 1, 1, 2, 48, 168, 1, 1, 4, 96, 11232, 20160, 1, 1, 2, 480, 86016, 24261120, 9999360, 1, 1, 6, 288, 1488000, 1321205760, 475566474240, 20158709760, 1, 1, 4, 2016, 1886976, 116064000000, 335522845163520, 84129611558952960, 163849992929280, 1

Offset: 0

Andrew Howroyd, Jul 08 2018

All rows are multiplicative.
Equivalently, the number of invertible n X n matrices mod k.
Also, for k prime (but not higher prime powers) the number of nonsingular n X n matrices over GF(k).
For k >= 2, n! divides T(n,k) since the subgroup of GL(n,k) consisting of all permutation matrices is isomorphic to S_n (the n-th symmetric group). Note that a permutation matrix is an orthogonal matrix, hence having determinant +-1. - Jianing Song, Oct 29 2022

			Array begins:
=================================================================
n\k| 1       2         3          4             5           6
---+-------------------------------------------------------------
0  | 1       1         1          1            1            1 ...
1  | 1       1         2          2            4            2 ...
2  | 1       6        48         96          480          288 ...
3  | 1     168     11232      86016      1488000      1886976 ...
4  | 1   20160  24261120 1321205760 116064000000 489104179200 ...
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 A000252, A064767, A305186.
Columns k=2..7 are A002884, A053290, A065128, A053292, A065498, A053293.
Cf. A053291 (GF(4)), A052496 (GF(8)), A052497 (GF(9)).
Cf. A316623.

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

T[, 1] = T[0, ] = 1; T[n_, k_] := T[n, k] = Module[{f = FactorInteger[k], p, e}, If[Length[f] == 1, {p, e} = f[[1]]; (p^e)^(n^2)* Product[(1 - 1/p^j), {j, 1, n}], Times @@ (T[n, Power @@ #]& /@ f)]];
Table[T[n - k + 1, k], {n, 0, 8}, {k, n + 1, 1, -1}] // Flatten (* Jean-François Alcover, Jul 25 2019 *)

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

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

A011787 Number of n X n matrices over Z_4 whose determinant is 1.

Original entry on oeis.org

1, 48, 43008, 660602880, 167761422581760, 692647993190048071680, 46119672943810238272430407680, 49327181820263339694887026906732953600, 845779385319367513935228608424336170436211507200, 232259029997997019092189603223864660200729932409175906713600

Offset: 1

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

nonn

Cf. A048651, A065128.

a[n_] := 2^(2*n^2-1) * Product[1 - 1/2^k, {k, 1, n}]; Array[a, 10] (* Amiram Eldar, Jul 12 2025 *)

a(n) = 2^(2*n^2-1) * prod(k = 1, n, 1-1/2^k); \\ Amiram Eldar, Jul 12 2025

a(n) = A065128(n)/2.

a(n) ~ c * 4^(n^2), where c = A048651 / 2. - Amiram Eldar, Jul 12 2025

More terms from Max Alekseyev, Jan 22 2012

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

A069580 Number of noninvertible n X n matrices mod 4 (i.e., over the ring Z_4).

Original entry on oeis.org

2, 160, 176128, 2973761536, 790377061679104, 3337070496489549070336, 224673304169436873829314985984, 241628003280411784073600553618302304256, 4154447778684876644944282114016459737751307157504, 1142419984262996237357582885893433282120743128964441021874176

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), Apr 18 2002

nonn

Cf. A011787, A048651, A065128.

a[n_] := 4^(n^2) * (1 - Product[1 - 1/2^k, {k, 1, n}]); Array[a, 10] (* Amiram Eldar, Jul 12 2025 *)

a(n) = 4^(n^2) * (1 - prod(k = 1, n, 1-1/2^k)); \\ Amiram Eldar, Jul 12 2025

a(n) = 4^(n^2) - A065128(n).

a(n) ~ c * 4^(n^2), where c = 1 - A048651. - Amiram Eldar, Jul 12 2025

More terms from Max Alekseyev, Jan 23 2012

cp's OEIS Frontend

A316622 Array read by antidiagonals: T(n,k) is the order of the group GL(n,Z_k).

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A011787 Number of n X n matrices over Z_4 whose determinant is 1.

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

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A069580 Number of noninvertible n X n matrices mod 4 (i.e., over the ring Z_4).

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