A064767 -id:A064767 - cp's OEIS frontend

A000252 Number of invertible 2 X 2 matrices mod n.

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

A000056 Order of the group SL(2,Z_n).

Original entry on oeis.org

1, 6, 24, 48, 120, 144, 336, 384, 648, 720, 1320, 1152, 2184, 2016, 2880, 3072, 4896, 3888, 6840, 5760, 8064, 7920, 12144, 9216, 15000, 13104, 17496, 16128, 24360, 17280, 29760, 24576, 31680, 29376, 40320, 31104, 50616, 41040, 52416, 46080, 68880, 48384, 79464

Offset: 1

N. J. A. Sloane

The number of equivalence classes of matrices modulo n of integer matrices with determinant 1 modulo n. - Michael Somos, Mar 20 2004
24 | a(n) if n > 2. - Michael Somos, Nov 15 2011
A divisibility sequence, that is, a(n) divides a(n*m) for all positive integers n and m. - Michael Somos, Jan 01 2017
The group SL(2,Z_2) is isomorphic to the symmetric group S_3. - Bernard Schott, Mar 15 2020
a(n) = [SL_2(Z) : Gamma(n)], index of the principal congruence subgroup of the special linear group over integers. - Andrey Zabolotskiy, Feb 14 2025

			G.f. = x + 6*x^2 + 24*x^3 + 48*x^4 + 120*x^5 + 144*x^6 + 336*x^7 +384*x^8 + ...
a(2) = 6 because [0, 1; 1, 0], [0, 1; 1, 1], [1, 0; 0, 1], [1, 0; 1, 1], [1, 1; 0, 1], [1, 1; 1, 0] are the six matrices modulo 2 with determinant 1 modulo 2.

T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 46.
B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 75.

T. D. Noe, Table of n, a(n) for n = 1..1000
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
Wikipedia, Congruence subgroup.
Index to divisibility sequences
Index entries for sequences related to groups

Cf. A001766.
Row n=2 of A316623.
Row sums of A316564.
Cf. A000252 (GL(2,Z_n)), A064767 (GL(3,Z_n)), A305186 (GL(4,Z_n)).
Cf. A011785 (SL(3,Z_n)), A011786 (SL(4,Z_n)).
Cf. A007434 ([SL_2(Z) : Gamma_1(n)]), A001615 ([SL_2(Z) : Gamma_0(n)]).

proc(n) local b,d: b := n^3: for d from 1 to n do if irem(n,d) = 0 and isprime(d) then b := b*(1-d^(-2)): fi: od: RETURN(b): end:

(* From Olivier Gérard, Aug 15 1997: (Start) *)
Table[ Fold[ If[ Mod[ n, #2 ]==0 && PrimeQ[ #2 ], #1*(1-1/#2^2), #1 ]&, n^3, Range[ n ] ], {n, 1, 35} ]
Table[ n^3 Times@@(1-1/Select[ Range[ 1, n ], (Mod[ n, #1 ]==0&&PrimeQ[ #1 ])& ]^2), {n, 1, 35} ]  (* End *)
a[ n_] := If[ n<1, 0, n Sum[ d^2 MoebiusMu[ n/d ], {d, Divisors @ n}]]; (* Michael Somos, Nov 15 2011 *)
Table[ n DirichletConvolve[ MoebiusMu[m], m^2, m, n], {n, 1, 35}] (* Li Han, Mar 15 2020 *)
a[n_] := #.RotateLeft[#] & @ Sort[Mod[ Outer[Times, Range[n], Range[n]], n] // Flatten // Tally][[;; , 2]]
Table[a[n], {n, 1, 35}] (* Li Han, Mar 15 2020 *)

{a(n) = if( n<1, 0, n * sumdiv(n, d, d^2 * moebius(n / d)))}; /* Michael Somos, Mar 05 2008 */

from math import prod
from sympy import factorint
def A000056(n): return prod((p+1)*(p-1)*p**(3*e-2) for p,e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

Multiplicative with a(p^e) = (p^2 - 1)*p^(3e-2). - David W. Wilson, Aug 01 2001
a(n) = A000252(n)/phi(n), where phi is Euler totient function (cf. A000010). - Vladeta Jovovic, Oct 30 2001
a(n) = n*Sum_{d|n} d^2*mu(n/d) = n*A007434(n) where A007434 is the Jordan function J_2(n). - Benoit Cloitre, May 03 2003
a(n) = A007434(n^2)/n. - Enrique Pérez Herrero, Sep 14 2010
a(n) = A007434(n^3)/n^3. - Enrique Pérez Herrero, Dec 19 2010
Dirichlet g.f. zeta(s-3)/zeta(s-1). - R. J. Mathar, Feb 27 2011
A046970(n) divides a(n). - R. J. Mathar, Mar 30 2011
Sum_{k=1..n} a(k) ~ n^4 / (4*Zeta(3)). - Vaclav Kotesovec, Jan 30 2019
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^2 / ((p-1)^2 * (p+1) * (p^2 + p + 1))) = 1.258448350408311046314826069717731136828991478925039589864338603650639811... - Vaclav Kotesovec, Sep 19 2020

More terms from Vaclav Kotesovec, Sep 19 2020

A064969 Number of cyclic subgroups 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, 8, 14, 36, 32, 112, 58, 148, 131, 256, 134, 504, 184, 464, 448, 596, 308, 1048, 382, 1152, 812, 1072, 554, 2072, 807, 1472, 1184, 2088, 872, 3584, 994, 2388, 1876, 2464, 1856, 4716, 1408, 3056, 2576, 4736, 1724, 6496, 1894, 4824, 4192, 4432, 2258, 8344

Offset: 1

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Oct 30 2001

Inverse Moebius transform of A160889. - Vladeta Jovovic, Nov 21 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
László Tóth, Menon's identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110 and arXiv, arXiv:1103.5861 [math.NT], 2011.
László Tóth, On the number of cyclic subgroups of a finite abelian group, arXiv:1203.6201 [math.GR], 2012.

Cf. A000010, A002117, A064803, A060648, A060724, A064767, A160889, A280184, A330595, A344219, A344302, A344303.

with(numtheory):
# define Jordan totient function J(r,n)
J(r,n) := add(d^r*mobius(n/d), d in divisors(n)):
seq(add(J(3,d)/phi(d), d in divisors(n)), n = 1..50); # Peter Bala, Jan 23 2024

a[n_] := Sum[EulerPhi[i] EulerPhi[j] (EulerPhi[k] / EulerPhi[LCM[i, j, k]]), {i, Divisors[n]}, {j, Divisors[n]}, {k, Divisors[n]}];
Array[a, 48] (* Jean-François Alcover, Dec 13 2018, after Vladeta Jovovic *)
f[p_, e_] := 1 + (p^2 + p + 1)*((p^(2*e) - 1)/(p^2 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 15 2022 *)

a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, eulerphi(i)*eulerphi(j)*eulerphi(k)/eulerphi(lcm(lcm(i, j), k))))); \\ Michel Marcus, Dec 14 2018

a160889(n) = sumdiv(n, d, moebius(n/d)*d^3)/eulerphi(n);
a(n) = sumdiv(n, d, a160889(d)); \\ Seiichi Manyama, May 12 2021

a(n) = Sum_{i|n, j|n, k|n} phi(i)*phi(j)*phi(k)/phi(lcm(i, j, k)), where phi is Euler totient function (cf. A000010).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + (p^2 + p + 1)*((p^(2*e) - 1)/(p^2 - 1)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/3) * Product_{p prime} (1 + 1/p^2 + 1/p^3) = A002117 * A330595 / 3 = 0.700772... . (End)
a(n) = Sum_{d divides n} J_3(d)/phi(d) = Sum_{1 <= i, j, k <= n} 1/phi(n/gcd(i,j,k,n)), where the Jordan totient function J_3(n) = A059376(n). - Peter Bala, Jan 23 2024

Formula and more terms from Vladeta Jovovic, Oct 30 2001

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

Original entry on oeis.org

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.

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

A011785 Number of 3 X 3 matrices whose determinant is 1 mod n.

Original entry on oeis.org

1, 168, 5616, 43008, 372000, 943488, 5630688, 11010048, 36846576, 62496000, 212427600, 241532928, 810534816, 945955584, 2089152000, 2818572288, 6950204928, 6190224768, 16934047920, 15998976000, 31621943808, 35687836800

Offset: 1

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

Order of the group SL(3,Z_n). For n > 2, a(n) is divisible by 48. - 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.

Cf. A000010, A000578, A003557, A003800, A046970, A063453.
Cf. A000056 (SL(2,Z_n)), A011786 (SL(4,Z_n)).
Cf. A000252 (GL(2,Z_n)), A064767 (GL(3,Z_n)), A305186 (GL(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]])/EulerPhi[n]; a[1] = 1; Array[a, 30] (* 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)))/eulerphi(n); \\ Michel Marcus, Jun 30 2015

from math import prod
from sympy import factorint
def A011785(n): return prod(p**((e<<3)-5)*(p**2*(p*(p-1)*(p+1)-1)+1) for p,e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

Multiplicative with a(p^e) = p^(8*e-5)*(p^3 - 1)*(p^2 - 1). - Vladeta Jovovic, Nov 18 2001
For a formula see A064767.
a(n) = A046970(n)*A063453(n)*A000578(n)*A003557(n)^5. - R. J. Mathar, Mar 30 2011
a(n) = A064767(n)/phi(n). - Jianing Song, Nov 24 2018
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^5/((p-1)^3 * (p+1)^2 * (p^2 + p + 1) * (p^6 + p^4 + p^2 + 1))) = 1.0061577672748872278355775942508642214184417621389767880397578015151659965... - Vaclav Kotesovec, Sep 19 2020
Sum_{k=1..n} a(k) ~ c * n^9, where c = (1/9) * Product_{p prime} (1 - (p^3 + p^2 -1)/p^6) = 0.08630488937... . - Amiram Eldar, Oct 23 2022

More terms from John W. Layman, Feb 16 2001

Further terms from Vladeta Jovovic, Oct 29 2001

A011786 Number of 4 X 4 matrices whose determinant is 1 mod n.

Original entry on oeis.org

1, 20160, 12130560, 660602880, 29016000000, 244552089600, 4635182361600, 21646635171840, 174060277297920, 584962560000000, 4139330225184000, 8013482872012800, 50858076935877120, 93445276409856000, 351980328960000000, 709316941310853120, 2851903720876769280

Offset: 1

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

Order of the group SL(4,Z_n). For n > 2, a(n) is divisible by 11520. - Jianing Song, Nov 24 2018

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000056 (SL(2,Z_n)), A011785 (SL(3,Z_n)).
Cf. A000252 (GL(2,Z_n)), A064767 (GL(3,Z_n)), A305186 (GL(4,Z_n)).
Cf. A000010.

f[p_, e_] := (1 - 1/p^4)*(1 - 1/p^3)*(1 - 1/p^2); a[1] = 1; a[n_] := n^15 * Times @@ f @@@ FactorInteger[n]; Array[a, 17] (* Amiram Eldar, Oct 23 2022 *)

a(n) = f = factor(n); n^16/eulerphi(n) * prod(i=1, #f~, (1-1/f[i,1]^4)*(1-1/f[i,1]^3)*(1-1/f[i,1]^2)*(1-1/f[i,1])); \\ Michel Marcus, Sep 02 2013

from math import prod
from sympy import factorint
def A011786(n): return prod(p**(3*(5*e-3))*(p**2*(p*(p*(p*(p*(p*(p-1)*(p+1)-1)-1)+1)+1)+1)-1) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

a(n) = (n^16/phi(n))*Product_{primes p dividing n} ((1 - 1/p^4)*(1 - 1/p^3)*(1 - 1/p^2)*(1 - 1/p)). Multiplicative with a(p^e) = p^(15*e-9)*(p^4 - 1)*(p^3 - 1)*(p^2 - 1). - Vladeta Jovovic, Nov 18 2001
a(n) = n^15*Product_{primes p dividing n} ((1 - 1/p^4)*(1 - 1/p^3)*(1 - 1/p^2)) = A305186(n)/phi(n). - Jianing Song, Nov 24 2018
Sum_{k=1..n} a(k) ~ c * n^16, where c = (1/16) * Product_{p prime} ((p^10 - p^7 - p^6 - p^5 + p^4 + p^3 + p^2 - 1)/p^10) = 0.04715136234... . - Amiram Eldar, Oct 23 2022

More terms from Vladeta Jovovic, Nov 18 2001

A137316 Array read by rows: T(n,k) is the number of automorphisms of the k-th group of order n, where the ordering is such that the rows are nondecreasing.

Original entry on oeis.org

1, 1, 2, 2, 6, 4, 2, 6, 6, 4, 8, 8, 24, 168, 6, 48, 4, 20, 10, 4, 12, 12, 12, 24, 12, 6, 42, 8, 8, 16, 16, 16, 32, 32, 32, 32, 48, 64, 96, 192, 192, 20160, 16, 6, 12, 48, 54, 432, 18, 8, 20, 24, 40, 40, 12, 42, 10, 110, 22, 8, 16, 16, 24, 24, 24, 24, 24, 24, 48, 48, 48, 48, 144, 336

Offset: 1

Benoit Jubin, Apr 06 2008, Apr 15 2008

The length of the n-th row is A000001(n).
The largest value of the n-th row is A059773(n).
The number phi(n) = A000010(n) appears in the n-th row.

			The table begins as follows:
   1
   1
   2
   2   6
   4
   2   6
   6
   4   8   8  24 168
   6  48
   4  20
  10
   4  12  12  12  24
  12
   6  42
The first row with two numbers corresponds to the two groups of order 4, the cyclic group Z_4 and the Klein group Z_2 x Z_2, whose automorphism groups are respectively the group (Z_4)^* = Z_2 and the symmetric group S_3.

D. MacHale and R. Sheehy, Finite groups with few automorphisms, Math. Proc. Roy. Irish Acad., 104A(2) (2004), 231--238.

Cf. A064767, A060249, A060817, A062771, A060249, A002618, A061350.

# GAP 4
Print("\n") ;
for o in [ 1 .. 33 ] do
    n := NumberSmallGroups(o) ;
    og := [] ;
    for i in [1 .. n] do
        g := SmallGroup(o,i) ;
        H := AutomorphismGroup(g) ;
        ho := Order(H) ;
        Add(og,ho) ;
    od;
    Sort(og) ;
    Print(og) ;
    Print("\n") ;
od; # R. J. Mathar, Jul 13 2013

A086768 Number of conjugacy classes in the group GL(3,Z_n).

Original entry on oeis.org

1, 6, 24, 60, 120, 144, 336, 536, 714, 720, 1320, 1440, 2184, 2016, 2880, 4528, 4896, 4284, 6840, 7200, 8064, 7920, 12144, 12864, 15540, 13104, 19908, 20160, 24360, 17280, 29760, 37216, 31680, 29376, 40320, 42840, 50616, 41040, 52416, 64320, 68880, 48384, 79464, 79200, 85680, 72864

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 02 2003

Cf. A064767, A062354.

[Nclasses(GeneralLinearGroup(3, ResidueClassRing(n))) : n in [2..50]]; // Robin Visser, Aug 06 2023

For a prime p : a(p) = p*(p^2 - 1).

More terms from Robin Visser, Aug 06 2023

A115222 Number of 3 X 3 matrices over Z(n) having nonzero determinant.

Original entry on oeis.org

0, 168, 11232, 161280, 1488000, 7170552, 33784128, 106659840, 327525120, 839997000, 2124276000, 4307378688, 9726417792, 18401146008, 34512588000

Offset: 1

T. D. Noe, Jan 16 2006

nonn

See A064767 for the number of invertible matrices.

Table[cnt=0; Do[m={{a, b, c}, {d, e, f}, {g, h, i}}; If[Det[m, Modulus->n]>0, cnt++ ], {a, 0, n-1}, {b, 0, n-1}, {c, 0, n-1}, {d, 0, n-1}, {e, 0, n-1}, {f, 0, n-1}, {g, 0, n-1}, {h, 0, n-1}, {i, 0, n-1}]; cnt, {n, 2, 7}]

a(n)=n^9-A115221(n). For prime n, a(n)=A064767(n)=n^3 (n-1)^3 (n+1) (n^2+n+1)

cp's OEIS Frontend

A000252 Number of invertible 2 X 2 matrices mod n.

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A000056 Order of the group SL(2,Z_n).

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A064969 Number of cyclic subgroups 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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A316622 Array read by antidiagonals: T(n,k) is the order of the group GL(n,Z_k).

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A305186 Number of invertible 4 X 4 matrices mod n.

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A011785 Number of 3 X 3 matrices whose determinant is 1 mod n.

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