A002884 -id:A002884 - cp's OEIS frontend

A061350 Maximal size of Aut(G) where G is a finite Abelian group of order n.

1, 1, 2, 6, 4, 2, 6, 168, 48, 4, 10, 12, 12, 6, 8, 20160, 16, 48, 18, 24, 12, 10, 22, 336, 480, 12, 11232, 36, 28, 8, 30, 9999360, 20, 16, 24, 288, 36, 18, 24, 672, 40, 12, 42, 60, 192, 22, 46, 40320, 2016, 480, 32, 72, 52, 11232, 40, 1008, 36, 28, 58, 48, 60, 30, 288

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 07 2001

a(n) is multiplicative; if n = p^m is a prime power the maximal size of Aut(G) is attained by the elementary Abelian group G =(C_p)^m and then Aut(G) is GL(m,p) and a(n) = (p^m - 1)*(p^m - p)*...*(p^m - p^(m-1)). For general n the maximum will be for the direct product of the (C_p)^m over the prime powers dividing n and then the automorphism group is the direct product of the GL(m,p).

Equivalently, maximal size of Aut(G) where G is a nilpotent group of order n. - Eric M. Schmidt, Feb 27 2013

T. D. Noe, Table of n, a(n) for n=1..1024

Cf. A059773, A002884, A053290, A053292, A053293.

A061350 := proc(n) local ans, i, j; ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*(mul(ifactors(n)[2][i][1]^ifactors(n)[2][i][2] - ifactors(n)[2][i][1]^(j - 1), j = 1..ifactors(n)[2][i][2])): od: RETURN(ans) end:

a[p_?PrimeQ] := p-1; a[1] = 1; a[n_] := Times @@ (Product[#[[1]]^#[[2]] - #[[1]]^k, {k, 0, #[[2]]-1}]& /@ FactorInteger[n]); Table[a[n], {n, 1, 63}] (* Jean-François Alcover, May 21 2012, after Maple *)

More terms from Vladeta Jovovic, Jun 12 2001

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

A270880 Triangle read by rows: T(n,m) is the number of direct-sum decompositions of a finite vector space of dimension n with m blocks over GF(2).

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 28, 28, 0, 1, 400, 1680, 840, 0, 1, 10416, 168640, 277760, 83328, 0, 1, 525792, 36053248, 159989760, 139991040, 27998208, 0, 1, 51116992, 17811244032, 209056841728, 419919790080, 227569434624, 32509919232

Offset: 0

Michel Marcus, Mar 25 2016

			Triangle begins:
1;
0, 1;
0, 1, 3;
0, 1, 28, 28;
0, 1, 400, 1680, 840;
0, 1, 10416, 168640, 277760, 83328;
...

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
David Ellerman, The number of direct-sum decompositions of a finite vector space, arXiv:1603.07619 [math.CO], 2016.
David Ellerman, The Quantum Logic of Direct-Sum Decompositions, arXiv preprint arXiv:1604.01087 [quant-ph], 2016. See Section 7.5.

Cf. A053601 (right diagonal), A270881 (row sums), A270882.

g[n_] := q^Binomial[n, 2] *FunctionExpand[QFactorial[n, q]]*(q - 1)^n /. q -> 2;Table[Table[Total[Map[g[n]/Apply[Times, g[#]]/Apply[Times, Table[Count[#, i], {i, 1, n}]!] &,IntegerPartitions[n, {m}]]], {m, 1, n}], {n, 1, 6}] // Grid (* Geoffrey Critzer, May 18 2017 *)

T(n,m) = Sum_ g(n)/(g(n_1)*g(n_2)***g(n_m))/(a_1!*a_2!***a_n!) where the sum is over all partitions of n into m parts and a_1,a_2,...,a_n is the part count signature of the partition and g(n) = A002884(n). - Geoffrey Critzer, May 18 2017 (after formula given in first Ellerman link above).

A289538 Expected dimension of the null space of a random linear operator on an n-dimensional vector space over the field with two elements as n -> infinity.

Original entry on oeis.org

8, 5, 0, 1, 7, 9, 8, 3, 0, 8, 7, 3, 9, 7, 9, 3, 3, 2, 8, 7, 6, 0, 6, 3, 2, 8, 1, 4, 9, 3, 5, 9, 1, 8, 7, 8, 8, 4, 0, 4, 2, 6, 7, 2, 5, 9, 7, 3, 2, 0, 2, 7, 2, 5, 9, 8, 7, 3, 5, 8, 0, 5, 2, 5, 5, 6, 3, 0, 9, 5, 9, 4, 1, 1, 8, 3, 3, 1, 3, 4, 4, 3, 6, 3, 0, 4, 1, 0, 6, 7, 0, 8, 8, 5, 9, 3, 5, 6, 5, 8

Offset: 0

Geoffrey Critzer, Jul 10 2017

More precisely, let X:L(V) -> {0,1,2,...,n} be the random variable that assigns to each linear operator T on n-dimensional vector space V over F_2, the integer j in {0,1,2,...,n} such that the dimension of the null space of T = j. Then E(X) = 0.850179183...

Cf. A002884, A182176, A289537, A289539, A289541, A289542.

nn = 300; q := 2;A[x_] := Sum[1/(FunctionExpand[QFactorial[j, q]] (q - 1)^j q^Binomial[j, 2]) Product[1 - 1/q^i, {i, j + 1, \[Infinity]}] x^j, {j, 0, nn}];RealDigits[
  N[Normal[Series[D[A[x], x] /. x -> 1, {x, 0, nn}]], 100]][[1]]

Let A(x) = Sum_{n>=0} Product_{i>=n+1} (1-1/2^i)*x^n/A002884(n). Then A'(1) = 0.85017983...

A335000 Orders of the groups PSL(m,q) in increasing order as q runs through the prime powers (with repetitions).

Original entry on oeis.org

6, 12, 60, 60, 168, 168, 360, 504, 660, 1092, 2448, 3420, 4080, 5616, 6072, 7800, 9828, 12180, 14880, 20160, 20160, 25308, 32736, 34440, 39732, 51888, 58800, 74412, 102660, 113460, 150348, 178920, 194472, 246480, 262080, 265680, 285852, 352440, 372000, 456288, 515100, 546312

Offset: 1

Michel Marcus, May 19 2020

nonn

60 is the order of PSL(2,4) and of PSL(2,5).
168 is the order of PSL(2,7) and of PSL(3,2).
20160 is the order of PSL(4,2) and of PSL(3,4).
Other repetitions > 20160 for PSL(m,q) groups are not known.
See A334884 and A334994 for variations of this sequence.

			a(5) = #PSL(2,7) = (7^2-1)*7/gcd(2,6) = 168, and,
a(6) = #PSL(3,2) = (2^3-1)*(2^3-2)*2^2/gcd(3,1) = 168.

Wikipedia, Projective linear group.
Index entries for sequences related to groups.

Cf. A002884 \ {1} (PSL(n,2)), A117762 (PSL(2, prime(n))).

Cf. A334884 (another case with repetitions), A334994 (without repetitions).

#PSL(m,q) = (Product_{j=0..m-2} (q^m - q^j)) * q^(m-1) / gcd(m,q-1). - Bernard Schott, May 19 2020

A346082 Number of cyclic n X n matrices over GF(2).

Original entry on oeis.org

1, 2, 14, 412, 50832, 25517184, 51759986688, 422000664182784, 13794938575436906496, 1805965390215106718072832, 946278871976706458877777936384, 1983897413727786229545246093886881792, 16639646499680599124923569106989157705580544, 558292116984541859085729903695019486031085083557888

Offset: 0

Geoffrey Critzer, Jul 04 2021

nonn

An n X n matrix A is cyclic if there is a vector v in GF(2)^n such that {A^i(v) : i>=0} spans GF(2)^n. Equivalently if the characteristic polynomial of A is equal to the minimal polynomial.

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

Cf. A001037, A002884.

Main diagonal of A347010.

nn = 13; A001037 = Table[1/n Sum[MoebiusMu[n/d] 2^d, {d, Divisors[n]}], {n, 1, nn}]; Table[Product[2^n - 2^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[
  Series[Product[(1 + 2^i x^i/((2^i - 1) (2^i - x^i)))^ A001037[[i]], {i, 1, nn}], {x, 0, nn}], x]

Sum_{n>=0} a(n) x^n/A002884(n) = Product_{i>=1} (1 + x^i/((2^i-1)(1-x/2)^i))^A001037(i).

A348015 Number of periodic n X n matrices over GF(2).

Original entry on oeis.org

1, 2, 13, 365, 43801, 21725297, 43798198753, 355991759464385, 11619571028917526401, 1520025803718875133673217, 796153035368657542014822907393, 1668838669721233396228446711227874305, 13995815633937307151473642050515241531340801

Offset: 0

Geoffrey Critzer, Sep 24 2021

nonn

Here, T is a periodic matrix if T = T^k for some k > 1. T is periodic iff image(T) = image(T^2) iff x^2 does not divide the minimal polynomial of T.

			a(2) = 13 because there are 16 2 X 2 matrices over GF(2) and all are recurrent except for {{0,0},{1,0}}, {{0,1},{0,0}}, and {{1,1},{1,1}}.  16-3 = 13.

Alois P. Heinz, Table of n, a(n) for n = 0..57
Geoffrey Critzer, Enumeration of matrices over finite fields
Eric Weisstein's World of Mathematics, Periodic Matrix

Cf. A002884.

Row sums of A348622.

b:= proc(n) option remember; mul(2^n-2^i, i=0..n-1) end:
a:= n-> add(b(n)/b(n-k), k=0..n):
seq(a(n), n=0..14);  # Alois P. Heinz, Oct 30 2021

nn=12; q = 2; b[p_, i_] := Count[p, i];
s[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^(s[p, i] deg) - q^((s[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1,Total[p]}]; \[Nu] =Table[1/n Sum[MoebiusMu[n/m] q^m, {m, Divisors[n]}], {n, 1,nn}];l[greatestpart_] := Level[Table[IntegerPartitions[n, {0, n}, Range[greatestpart]], {n, 0, nn}], {2}];
g1[u_, v_, deg_] :=Total[Map[v u^(deg Total[#])/aut[deg, #] &, l[1]]];
g2[u_, v_, deg_] :=Total[Map[v u^(deg Total[#])/aut[deg, #] &, l[nn]]];
Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[
  Series[g1[u, 1, 1] g2[u, 1, 1] Product[g2[u, 1, d]^\[Nu][[d]], {d, 2, nn}] , {u, 0, nn}], u]

a(n) = Sum_{d=0...n} A002884(n)/A002884(n-d). - Geoffrey Critzer, Oct 30 2021
Sum_{n>=0} a(n)u^n/A002884(n) = E(u)/(1-u) where E(u) = Sum_{n>=0} u^n/A002884(n). - Geoffrey Critzer, Oct 30 2021
Limit_{n->infinity} a(n)/2^(n^2) = (Product_{r>=1} (1-1/2^r)) * Sum_{n>=0} 1/A002884(n) = 0.62744086981206237307... . - Geoffrey Critzer, Oct 30 2021
Sum_{n>=0} a(n)*x^n/B(n) = e(x)*g(x) where e(x) = Sum_{n>=0} x^n/B(n), g(x) = Sum_{n>=0} Product_{i=0..n-1} (q^n-q^i)*x^n/B(n), B(n) = Product_{i=0..n-1} (q^n-q^i)/(q-1)^n and q=2. - Geoffrey Critzer, Jan 03 2025

Data terms for n >= 3 corrected by Geoffrey Critzer, Oct 30 2021

Title improved by Geoffrey Critzer, Sep 16 2022

A062240 Number of subgroups of Chevalley group A_n(2) (the group of nonsingular n X n matrices over GF(2) ).

Original entry on oeis.org

1, 6, 179, 48337, 54091780, 922300149178

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 30 2001

Cf. A002884.

[&+[s`length : s in Subgroups(GL(n, 2))] : n in [1..6]];  // Robin Visser, Aug 14 2023

a(5)-a(6) from Robin Visser, Aug 14 2023

A086699 Number of n X n matrices over GF(2) with rank n-1.

Original entry on oeis.org

1, 9, 294, 37800, 19373760, 39687459840, 325139829719040, 10654345790226432000, 1396491759480328106803200, 732164571206732295657278668800, 1535460761275478347250381697633484800, 12880379193826999985837000446453418557440000

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 28 2003

nonn

a(n)/2^(n^2) is the probability that a random linear operator T on an n dimensional vector space over the field with two elements is such that the dimension of the range of T equals n-1. This probability is Product{j>=2} 1 - 1/2^j which is 2 times the probability that the dimension of the range of T equals n. Cf. A048651. - Geoffrey Critzer, Jun 28 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..58

Cf. A002884, A060867.

Table[Product[(q^n - q^i)^2/(q^(n - 1) - q^i), {i, 0, n - 2}] /.  q -> 2, {n, 0, 15}] (* Geoffrey Critzer, Jun 28 2017 *)

a(n) = prod(j=0, n-2, (2^n - 2^j)^2 / (2^(n-1)- 2^j)); \\ Michel Marcus, Jun 28 2017

for n>=2 : a(n) = product j=0...n-2 (2^n - 2^j)^2 / (2^(n-1)- 2^j).

More terms from David Wasserman, Mar 28 2005

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.

cp's OEIS Frontend

A061350 Maximal size of Aut(G) where G is a finite Abelian group of order n.

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

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A270880 Triangle read by rows: T(n,m) is the number of direct-sum decompositions of a finite vector space of dimension n with m blocks over GF(2).

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A289538 Expected dimension of the null space of a random linear operator on an n-dimensional vector space over the field with two elements as n -> infinity.

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Formula

A335000 Orders of the groups PSL(m,q) in increasing order as q runs through the prime powers (with repetitions).

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Formula

A346082 Number of cyclic n X n matrices over GF(2).

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Mathematica

Formula

A348015 Number of periodic n X n matrices over GF(2).

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Maple

Mathematica

Formula

Extensions

A062240 Number of subgroups of Chevalley group A_n(2) (the group of nonsingular n X n matrices over GF(2) ).

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