A011785 -id:A011785 - 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

A064767 Order of automorphism group 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, 168, 11232, 86016, 1488000, 1886976, 33784128, 44040192, 221079456, 249984000, 2124276000, 966131712, 9726417792, 5675733504, 16713216000, 22548578304, 111203278848, 37141348608, 304812862560, 127991808000

Offset: 1

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Oct 24 2001

Also number of 3 X 3 invertible matrices over the ring Z/nZ. - Max Alekseyev, Nov 02 2007

Order of the group GL(3,Z_n). For n > 2, a(n) is divisible by 96. - 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.
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.
Index entries for sequences related to groups.

Row n=3 of A316622.
Cf. A000010, A064803.
Cf. A000252 (GL(2,Z_n)), A305186 (GL(4,Z_n)).
Cf. A000056 (SL(2,Z_n)), A011785 (SL(3,Z_n)), A011786 (SL(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]]; a[1] = 1; Array[a, 20] (* 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))); \\ Michel Marcus, Jun 30 2015

a(n,f=factor(n))=prod(i=1,#f~, ((1 - 1/f[i,1]^3)*(1 - 1/f[i,1]^2)*(1 - 1/f[i,1])))*n^9 \\ Charles R Greathouse IV, Mar 04 2025

from math import prod
from sympy import factorint
def A064767(n): return prod(p**(3*(3*e-2))*(p*(p*(p**2*(p*(p-1)-1)+1)+1)-1) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

a(n) = phi(n)*A011785(n). - Vladeta Jovovic, Oct 29 2001
a(n) = n^9*Product_{primes p dividing n} ((1 - 1/p^3)*(1 - 1/p^2)*(1 - 1/p)). This also gives a formula for A011785.
Multiplicative with a(p^e) = p^(9*e-6)*(p^3 - 1)*(p^2 - 1)*(p - 1). - Vladeta Jovovic, Nov 18 2001
Sum_{k=1..n} a(k) ~ c * n^10, where c = (1/10) * Product_{p prime} ((p^7 - p^5 - p^4 + p^2 + p - 1)/p^7) =  0.05123382571... . - Amiram Eldar, Oct 23 2022

More terms from Vladeta Jovovic, Nov 18 2001

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

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

A115224 Number of 3 X 3 symmetric matrices over Z(n) having determinant 1.

Original entry on oeis.org

1, 28, 234, 896, 3100, 6552, 16758, 28672, 56862, 86800, 160930, 209664, 371124, 469224, 725400, 917504, 1419568, 1592136, 2475738, 2777600, 3921372, 4506040, 6435814, 6709248, 9687500, 10391472, 13817466, 15015168, 20510308, 20311200, 28628190, 29360128, 37657620

Offset: 1

T. D. Noe, Jan 16 2006

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..4885 from Enrique Pérez Herrero)

Cf. A000056 (order of the group SL(2, Z_n)), A011785 (number of 3 X 3 matrices whose determinant is 1 mod n, i.e. order of SL(3, Z_n)).

Table[cnt=0; Do[m={{a, b, c}, {b, d, e}, {c, e, f}}; If[Det[m, Modulus->n]==1, 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}]; cnt, {n, 2, 20}]
JordanTotient[n_,k_:1] := DivisorSum[n,#^k*MoebiusMu[n/# ]&]/;(n>0)&&IntegerQ[n]; A115224[n_IntegerQ] := JordanTotient[n^2,3]/n; Table[A115224[n], {n,100}] (* Enrique Pérez Herrero, Sep 14 2010 *)
f[p_, e_] := (p^3 - 1)*p^(5*e - 3); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 15 2020 *)

a(1)=1 because the matrix of all zeros has determinant 0, but 0=1 (mod 1).
For prime p, a(p) = (p^3-1)*p^2.
Multiplicative with a(p^e) = (p^3-1)*p^(5e-3).
a(n) = A011785(n)/A000056(n).
a(n) = A059376(n^2)/n. - Enrique Pérez Herrero, Sep 14 2010
a(n) = n^2*A059376(n). Dirichlet g.f.: zeta(s-5)/zeta(s-2). - R. J. Mathar, Feb 27 2011
Sum_{k=1..n} a(k) ~ 15*n^6 / Pi^4. - Vaclav Kotesovec, Feb 07 2019
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^3/(1 - p^3 - p^5 + p^8)) = 1.04172462829914219180789244796430293454403616906393417764614215669994022537... - Vaclav Kotesovec, Sep 20 2020

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

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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A064767 Order of automorphism group 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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A305186 Number of invertible 4 X 4 matrices mod n.

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

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A115224 Number of 3 X 3 symmetric matrices over Z(n) having determinant 1.

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A316623 Array read by antidiagonals: T(n,k) is the order of the group SL(n,Z_k).

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