A002820 -id:A002820 - 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

A005327 Certain subgraphs of a directed graph (inverse binomial transform of A005321).

Original entry on oeis.org

1, 0, 1, 6, 91, 2820, 177661, 22562946, 5753551231, 2940064679040, 3007686166657921, 6156733583148764286, 25211824022994189751171, 206510050572345408251841660, 3383254158526734823389921915781

Offset: 1

N. J. A. Sloane

nonn

q-Subfactorial for q=2. - Vladimir Reshetnikov, Sep 12 2016

T. L. Greenough, Enumeration of interval orders without duplicated holdings, Preprint, circa 1976.
T. L. Greenough and T. Lockman, Representation and enumeration of interval orders and semiorders, Ph.D. Thesis, Dartmouth, 1976.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..50
E. Andresen and K. Kjeldsen, On certain subgraphs of a complete transitively directed graph, Discrete Math. 14 (1976), no. 2, 103-119.
T. L. Greenough, Enumeration of interval orders without duplicated holdings, Preprint, circa 1976. [Annotated scanned copy]
Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Subfactorial, q-Factorial, q-Analog.

Cf. A002820, A005329, A005321.

p:=proc(n) if n=0 then 1 else product(2^i-1,i=1..n) fi end: a:=n->p(n-1)*sum((-1)^j/p(j),j=0..n-1): seq(a(n),n=1..17); # Emeric Deutsch, Jan 23 2005

a[1] = 1; a[n_] := a[n] = (2^(n-1)-1)*a[n-1] + (-1)^(n-1); Array[a, 15] (* Jean-François Alcover, Apr 05 2016, after Max Alekseyev *)
With[{q = 2}, Table[QFactorial[n, q] Sum[(-1)^k/QFactorial[k, q], {k, 0, n}], {n, 0, 20}]] (* Vladimir Reshetnikov, Sep 12 2016 *)

For n>1, a(n) = (2^(n-1)-1)*a(n-1) + (-1)^(n-1). - Max Alekseyev, Feb 23 2012
a(n) = p(n-1)*sum((-1)^j/p(j), j=0..n-1), where p(0) = 1, p(k) = product(2^i-1, i=1..k) for k>0. - Emeric Deutsch, Jan 23 2005
a(n) ~ A048651^2 * 2^(n*(n-1)/2). - Vaclav Kotesovec, Oct 09 2019

More terms from Max Alekseyev, Apr 27 2010

A218530 Partial sums of floor(n/11).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 171

Offset: 0

Philippe Deléham, Mar 27 2013

Apart from the initial zeros, the same as A008729.

			As square array:
..0....0....0....0....0....0....0....0....0....0....0
..1....2....3....4....5....6....7....8....9...10...11
.13...15...17...19...21...23...25...27...29...31...33
.36...39...42...45...48...51...54...57...60...63...66
.70...74...78...82...86...90...94...98..102..106..110
115..120..125..130..135..140..145..150..155..160..165
171..177..183..189..195..201..207..213..219..225..231
238..245..252..259..266..273..280..287..294..301..308
316..324..332..340..348..356..364..372..380..388..396
405..414..423..432..441..450..459..468..477..486..495
505..515..525..535..545..555..565..575..585..595..605
...

Cf. similar sequences: A000217, A002820, A130518, A130519, A130520, A174709, A174738, A118729, A218470, A131242.

a(11n)    = A051865(n).
a(11n+1)  = A180223(n).
a(11n+4)  = A022268(n).
a(11n+5)  = A022269(n).
a(11n+6)  = A254963(n)
a(11n+9)  = A211013(n).
a(11n+10) = A152740(n).
G.f.: x^11/((1-x)^2*(1-x^11)).

A051680 Number of n X n invertible matrices A over GF(3) such that A-I is invertible.

Original entry on oeis.org

1, 27, 6291, 13589289, 266377183929, 47123189360124723, 75095231825148137471259, 1077370264330489309698453375441, 139124702920688202983704723564457669361

Offset: 1

Vladeta Jovovic, Mar 17 2000

Cf. A002820.

a[n_] := a[n] = 3^(n-1)*((3^n-1)*a[n-1] + (-1)^n*3^((n-3)*n/2+1)); a[1] = 1; Table[a[n], {n, 1, 9}] (* Jean-François Alcover, Jan 12 2012, after formula *)

a(n) = 3^binomial(n,2)*b(n), with b(0)=1, b(n)=(3^n-1)*b(n-1)+(-1)^(n). - Vladeta Jovovic, Aug 20 2006
From Geoffrey Critzer, Oct 17 2021: (Start)
Sum_{n>=0} a(n)*u^n/A053290(n) = 1/(1-u)*Product_{r>=1} 1-u/3^r.
Limit_{n->inf} a(n)/3^(n^2) = (Product_{r>=1} 1-1/3^r)^2. (End)

A346209 Number of n X n matrices over GF(3) with no eigenvalues in GF(3), i.e., neither 0 nor 1 nor 2 is an eigenvalue.

Original entry on oeis.org

1, 0, 18, 3456, 7619508, 149200289280, 26394940582090344, 42062797470468915399168, 603463180651533072058654437264, 77927374189849689541269666899007713280, 90570450400853976077932766909301405665963077152

Offset: 0

Geoffrey Critzer, Jul 10 2021

nonn

Equivalently, a(n) is the number of n X n matrices over GF(3) whose characteristic polynomial has no linear factors.

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A002820, A051680, A027376, A053290.

nn = 10; q = 3; \[Nu] = Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}];Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[Product[Product[1/(1 - u^d/q^(r d)), {r, 1, \[Infinity]}]^\[Nu][[d]], {d, 2, nn}], {u, 0, nn}], u]

Sum_{n>=0} a(n)*x^n/A053290(n) = Product_{d>=2} (Product_{r>=1} 1/(1-x^d/3^(r*d)))^A027376(d).

A346381 Triangle read by rows. T(n,k) is the number of invertible n X n matrices over GF(2) such that the dimension of the eigenspace corresponding to eigenvalue 1 is k, 0 <= k <= n, n >= 0.

Original entry on oeis.org

1, 0, 1, 2, 3, 1, 48, 98, 21, 1, 5824, 11640, 2590, 105, 1, 2887680, 5775424, 1283400, 52390, 465, 1, 5821595648, 11643190272, 2587376064, 105607080, 938742, 1953, 1, 47317927329792, 94635854692352, 21030189917184, 858375102144, 7630000488, 15879318, 8001, 1

Offset: 0

Geoffrey Critzer, Jul 14 2021

			Triangle begins:
        1;
        0,       1;
        2,       3,      1;
       48,      98,     21,      1;
     5824,   11640,   2590,    105,   1;
  2887680, 5775424, 1283400, 52390, 465, 1;
  ...
T(2,0) = 2 because {{0, 1}, {1, 1}}, {{1, 1}, {1, 0}} do not have 1 as an eigenvalue.
T(2,1) = 3 because {{0, 1}, {1, 0}}, {{1, 0}, {1, 1}}, {{1, 1}, {0, 1}} have 1 as an eigenvalue with corresponding eigenspace of dimension 1.
T(2,2) = 1 because {{1, 0}, {0, 1}} fixes the entire space.

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A002820 (column k=0), A002884 (row sums).

nn = 15; q = 2; b[p_, i_] := Count[p, i];d[p_, i_] := Sum[j b[p, j], {j, 1, i}] + i Sum[b[p, j], {j, i + 1, Total[p]}]; aut[deg_, p_] := Product[Product[ q^(d[p, i] deg) - q^((d[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1,Total[p]}]; A001037 =
Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}];
g[u_, v_] := Total[Map[v^Length[#] u^Total[#]/aut[1, #] &,Level[Table[IntegerPartitions[n], {n, 0, nn}], {2}]]];Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[g[u, v] Product[Product[1/(1 - (u/q^r)^d), {r, 1, \[Infinity]}]^A001037[[d]], {d, 2, nn}], {u, 0, nn}], {u, v}] // Grid

For n>=1, Sum_{k=0..n} T(n,k)*2^k = 2*A002884(n). - Geoffrey Critzer, Jan 10 2025

A053060 Number of n X n invertible binary matrices A such that A^3+I is invertible.

Original entry on oeis.org

0, 0, 48, 4032, 1935360, 4022599680, 32585690382336, 1063349908766982144, 139112376066747546992640, 72863495247749516035746693120, 152731190537447000980973281829978112

Offset: 1

Vladeta Jovovic, Mar 18 2000

nonn

V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Cf. A002820, A053071.

A053071 Number of n X n invertible binary matrices A such that A^5+I is invertible.

Original entry on oeis.org

0, 2, 48, 4480, 2887680, 5373624320, 44196975083520, 1442855588252876800, 188570467779447305011200, 98800579402758985681259724800, 207089099087390763078860569942425600

Offset: 1

Vladeta Jovovic, Mar 18 2000

nonn

V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Cf. A002820, A051680, A053060.

A346201 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(2) such that the sum of the dimensions of their eigenspaces taken over all eigenvalues is k, 0 <= k <= n, n >= 0.

Original entry on oeis.org

1, 0, 2, 2, 6, 8, 48, 196, 210, 58, 5824, 23280, 27020, 8610, 802, 2887680, 11550848, 13756560, 4757260, 581250, 20834, 5821595648, 23286380544, 28097284992, 10075582800, 1369706604, 67874562, 1051586, 47317927329792, 189271709384704, 229853403924480, 83865929653632, 11957394226896, 668707460652, 14779207170, 102233986

Offset: 0

Geoffrey Critzer, Jul 16 2021

			        1;
        0,        2;
        2,        6,        8;
       48,      196,      210,      58;
     5824,    23280,    27020,    8610,    802;
  2887680, 11550848, 13756560, 4757260, 581250, 20834;

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A002820 (column k=0), A132186 (main diagonal), A002416 (row sums).

nn = 8; q = 2; b[p_, i_] := Count[p, i];d[p_, i_] :=  Sum[j b[p, j], {j, 1, i}] + i Sum[b[p, j], {j, i + 1, Total[p]}];aut[deg_, p_] :=  Product[Product[q^(d[p, i] deg) - q^((d[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1,Total[p]}]; A001037 =
Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}];g[u_, v_] :=
Total[Map[v^Length[#] u^Total[#]/aut[1, #] &,Level[Table[IntegerPartitions[n], {n, 0, nn}], {2}]]];Table[Take[(Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[g[u, v] g[u, v] Product[Product[1/(1 - (u/q^r)^d), {r, 1, \[Infinity]}]^A001037[[d]], {d, 2, nn}], {u, 0, nn}], {u, v}])[[n]],
   n], {n, 1, nn}] // Grid

A344886 a(n) is the smallest triangular number that is a multiple of the product of the members of the n-th pair of twin primes.

Original entry on oeis.org

15, 105, 2145, 11628, 94395, 370230, 1565565, 3265290, 13263825, 16689753, 44674878, 62434725, 129757995, 168095280, 190173753, 334822503, 411256860, 659371455, 784892010, 1176876870, 1822721253, 3871076055, 4333386060, 5670113295, 9245348190, 13148662530

Offset: 1

Ali Sada, Jun 01 2021

nonn

If we divide each a(n) by the two primes we get a sequence of the triangular numbers of (3 * A002820(n) - 1). If we take the differences between those triangular numbers we get A145061 + 1.

This is a subsequence of A011772, which is really the basic sequence here. - N. J. A. Sloane, Jul 06 2021

			15 is the smallest triangular number that is a multiple of 3 and 5, so, a(1) = 15.

Cf. A001359, A006512, A000217, A308344, A002820, A145061.

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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A005327 Certain subgraphs of a directed graph (inverse binomial transform of A005321).

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A218530 Partial sums of floor(n/11).

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A051680 Number of n X n invertible matrices A over GF(3) such that A-I is invertible.

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A346209 Number of n X n matrices over GF(3) with no eigenvalues in GF(3), i.e., neither 0 nor 1 nor 2 is an eigenvalue.

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A346381 Triangle read by rows. T(n,k) is the number of invertible n X n matrices over GF(2) such that the dimension of the eigenspace corresponding to eigenvalue 1 is k, 0 <= k <= n, n >= 0.

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A053060 Number of n X n invertible binary matrices A such that A^3+I is invertible.

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A053071 Number of n X n invertible binary matrices A such that A^5+I is invertible.

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A346201 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(2) such that the sum of the dimensions of their eigenspaces taken over all eigenvalues is k, 0 <= k <= n, n >= 0.

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