A000252 -id:A000252 - cp's OEIS frontend

A065501 Number of conjugacy classes in the group SL(2,Z_n) (see A000056).

1, 3, 7, 10, 9, 21, 11, 30, 25, 27, 15, 70, 17, 33, 63, 76, 21, 75, 23, 90, 77, 45, 27, 210, 49, 51, 79, 110, 33, 189, 35, 168, 105, 63, 99, 250, 41, 69, 119, 270, 45, 231, 47, 150, 225, 81, 51, 532, 81, 147, 147, 170, 57, 237, 135, 330, 161, 99, 63, 630, 65, 105, 275, 352, 153

Offset: 1

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 25 2001

Robin Visser, Table of n, a(n) for n = 1..500

Cf. A062354, A000056, A000252

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

For an odd prime p : a(p) = p + 4. - Robin Visser, Aug 06 2023

a(23) corrected and more terms from Robin Visser, Aug 06 2023

A181107 Triangle read by rows: T(n,k) is the number of 2 X 2 matrices over Z(n) having determinant congruent to k mod n, 1 <= n, 0 <= k <= n-1.

Original entry on oeis.org

1, 10, 6, 33, 24, 24, 88, 48, 72, 48, 145, 120, 120, 120, 120, 330, 144, 240, 198, 240, 144, 385, 336, 336, 336, 336, 336, 336, 736, 384, 576, 384, 672, 384, 576, 384, 945, 648, 648, 864, 648, 648, 864, 648, 648, 1450, 720, 1200, 720, 1200, 870, 1200, 720, 1200, 720

Offset: 1

Erdos Pal, Oct 03 2010

The n-th row is {T(n,0),T(n,1),...,T(n,n-1)}.
Let m denote the prime power p^e.
T(m,0) = A020478(m) = (p^(e+1) + p^e-1)*p^(2*e-1).
T(m,1) = A000056(m) = (p^2-1)*p^(3*e-2).
T(prime(n),1) = A127917(n).
Sum_{k=1..n-1} T(n,k) = A005353(n).
T(n,1) = n*A007434(n) for n>=1 because A000056(n) = n*Jordan_Function_J_2(n).
T(2^n,1) = A083233(n) = A164640(2n) for n>=1. Proof: a(n):=T(2^n,1); a(1)=6, a(n)=8*a(n-1); A083233(1)=6 and A083233(n) is a geometric series with ratio 8 (because of its g.f.), too; A164640 = {b(1)=1, b(2)=6, b(n)=8*b(n-2)}.
T(2^n,0) = A165148(n) for n>=0, because  2*T(2^n,0) = (3*2^n-1)*4^n.
T(2^e,2) = A003951(e) for 2 <= e. Proof: T(2^e,2) = 9*8^(e-1) is a series with ratio 8 and initial term 72, as A003951(2...inf) is.
Working with consecutive powers of a prime p, we need a definition (0 <= i < e):
N(p^e,i):=#{k: 0 < k < p^e, gcd(k,p^e) = p^i} = (p-1)*p^(e-1-i). We say that these k's belong to i (respect to p^e). Note that N(p^e,0) = EulerPhi(p^e), and if 0 < k < p^e then gcd(k,p^e) = gcd(k,p^(e+1)). Let T(p^e,[i]) denote the common value of T(p^e,k)'s, where k's belong to i (q.v.PROGRAM); for example, T(p^e,[0]) = T(p^e,1). The number of the 2 X 2 matrices over Z(p^e), T(p^e,0) + Sum_{i=0..e-1} T(p^e,[i])*N(p^e,i) = p^(4e) will be useful.
On the hexagon property: Let prime p be given and let T(p^e,[0]), T(p^e,[1]), T(p^e,[2]), ..., T(p^e,[e-2]), T(p^e,[e-1]) form the e-th row of a Pascal-like triangle, e>=1. Let denote X(r,s) an element of the triangle and its value T(p^r,[s]). Let positive integers a and b given, so that the entries A(m-a,n-b), B(m-a,n), C(m,n+a), D(m+b,n+a), E(m+b,n), F(m,n-b) of the triangle form a hexagon spaced around T(p^m,[n]); if a=b=1 then they surround it. If A*C*E = B*D*F, then we say that the triangle T(.,.) has the "hexagon property". (In the case of binomial coefficients X(r,s) = COMB(r,s), the "hexagon property" holds (see [Gupta]) and moreover gcd(A,C,E) = gcd(B,D,F) (see [Hitotumatu & Sato]).)
Corollary 2.2 in [Brent & McKay] says that, for the d X d matrices over Z(p^e), (mutatis mutandis) T_d(p^e,0) = K*(1-P(d+e-1)/P(e-1)) and T_d(p^e,[i]) = K*(q^e)*((1-q^d)/(1-q))*P(d+i-1)/P(i), where q=1/p, K=(p^e)^(d^2), P(t) = Product_{j=1..t} (1-q^j), P(0):=1. (For the case d=2, we have T(p^e,[i]) = (p+1)*(p^(i+1)-1)*p^(3*e-i-2).) Due to [Brent & McKay], it can be simply proved that for d X d matrices the "hexagon property" is true. The formulation implies an obvious generalization: For the entries A(r,u), B(r,v), C(s,w), D(t,w), E(t,v), F(s,u) of the T_d(.,.)-triangle, a hexagon-like property A*C*E = B*D*F holds. This is false in general for the COMB(.,.)-triangle.
Another (rotated-hexagon-like) property: for the entries A(m-b1,n), B(m-a1,n+c2), C(m+a2,n+c2), D(m+b2,n), E(m+a2,n-c1), F(m-a1,n-c1) of the T_d(.,.)-triangle, the property A*C*E = B*D*F holds, if and only if 2*(a1 + a2) = b1 + b2. This is also in general false for COMB(.,.)-triangle.

			From _Andrew Howroyd_, Jul 16 2018: (Start)
Triangle begins:
    1;
   10,   6;
   33,  24,  24;
   88,  48,  72,  48;
  145, 120, 120, 120, 120;
  330, 144, 240, 198, 240, 144;
  385, 336, 336, 336, 336, 336, 336;
  736, 384, 576, 384, 672, 384, 576, 384;
  945, 648, 648, 864, 648, 648, 864, 648, 648;
  ... (End)

Erdos Pal, Rows n=1..100 of triangle, flattened
Richard P. Brent and Brendan D. McKay, Determinants and ranks of random matrices over Z_m, Discrete Mathematics 66 (1987) pp. 35-49.
A. K. Gupta, Generalized hidden hexagon squares, The Fibonacci Quarterly, Vol 12, Number 1, Feb.1974, pp. 45-46.
S. Hitotumatu, D. Sato, Star of David theorem (I), The Fibonacci Quarterly, Vol 13, Number 1, Feb.1975, p. 70.

Column k=0 is A020478.
Column k=1 is A000056.
Row sums are A005353.
Cf. A000252, A127917.

  (* computing T(p^e,k) ; p=prime, 1<=e, 0<=k

S(p,e)={my(u=vector(p^e)); my(t=(p-1)*p^(e-1)); u[1] = p^e + e*t; for(j=1, p^e-1, u[j+1] = t*(1+valuation(j, p))); vector(#u, k, sum(j=0, #u-1, u[j + 1]*u[(j+k-1) % #u + 1]))}
T(n)={my(f=factor(n), v=vector(n,i,1)); for(i=1, #f~, my(r=S(f[i,1], f[i,2])); for(j=0, #v-1, v[j + 1] *= r[j % #r + 1])); v}
for(n=1, 10, print(T(n))); \\ Andrew Howroyd, Jul 16 2018

T(a*b,k) = T(a,(k mod a))*T(b,(k mod b)) if gcd(a,b) = 1.

Sum_{k=1..n-1, gcd(k,n)=1} T(n,k) = A000252(n). - Andrew Howroyd, Jul 16 2018

Terms a(24)-a(55) from b-file by Andrew Howroyd, Jul 16 2018

A327568 Exponent of the group GL(2, Z_n).

Original entry on oeis.org

1, 6, 24, 12, 120, 24, 336, 24, 72, 120, 1320, 24, 2184, 336, 120, 48, 4896, 72, 6840, 120, 336, 1320, 12144, 24, 600, 2184, 216, 336, 24360, 120, 29760, 96, 1320, 4896, 1680, 72, 50616, 6840, 2184, 120, 68880, 336, 79464, 1320, 360, 12144

Offset: 1

Jianing Song, Sep 17 2019

nonn

The exponent of a finite group G is the least positive integer k such that x^k = e for all x in G, where e is the identity of the group. That is to say, the exponent of a finite group G is the LCM of the orders of elements in G. Of course, the exponent divides the order of the group.

			GL(2, Z_2) is isomorphic to S_3, which has 1 identity element, 3 elements with order 2 and 2 elements with order 3, so a(2) = lcm(1, 2, 3) = 6.

Kenneth G. Hawes, Table of n, a(n) for n = 1..200
The Group Properties Wiki, Exponent of a group

Cf. A000252, A316565, A327569.

MatOrder(M)={my(id=matid(#M), k=1, N=M); while(N<>id, k++; N=N*M); k}
a(n)={my(m=1); for(a=0, n-1, for(b=0, n-1, for(c=0, n-1, for(d=0, n-1, my(M=Mod([a, b; c, d], n)); if(gcd(lift(matdet(M)), n)==1, m=lcm(m, MatOrder(M))))))); m} \\ Following Andrew Howroyd's program for A316565

If gcd(m, n) = 1 then a(m*n) = lcm(a(m), a(n)).

Conjecture: a(p^e) = (p^2-1)*p^e for primes p. If this is true, then 24 divides a(n) for n > 2.

A020479 Number of noninvertible 2 X 2 matrices over Z/nZ (determinant is a divisor of 0).

Original entry on oeis.org

10, 33, 160, 145, 1008, 385, 2560, 2673, 7120, 1441, 16128, 2353, 26320, 27585, 40960, 5185, 81648, 7201, 113920, 97713, 155056, 12673, 258048, 90625, 299728, 216513, 421120, 25201, 671760, 30721, 655360, 552321, 866320, 532945, 1306368, 51985

Offset: 2

David W. Wilson

nonn

Cf. A000252.

f[p_, e_] := (p - 1)^2*(p + 1)*p^(4*e - 3); a[n_] := n^4 - Times @@ f @@@ FactorInteger[n]; Array[a, 36, 2] (* Amiram Eldar, Aug 03 2024 *)

a(n) = {my(f = factor(n), p = f[,1], e=f[,2]); n^4 - prod(k = 1, #p, (p[k] - 1)^2*(p[k] + 1)*p[k]^(4*e[k] - 3));} \\ Amiram Eldar, Aug 03 2024

a(n) seems to be divisible by n. - Ralf Stephan, Sep 01 2003 [This is true and can be easily proven from the formula below and from the multiplicative formula for A000252(n). - Amiram Eldar, Aug 03 2024]

a(n) = n^4 - A000252(n). - T. D. Noe, Jan 16 2006

A065499 Noninvertible 3 X 3 matrices over Z_n.

Original entry on oeis.org

0, 344, 8451, 176128, 465125, 8190720, 6569479, 90177536, 166341033, 750016000, 233671691, 4193648640, 878081581, 14985313280, 21730143375, 46170898432, 7384597649, 161217941760, 17874835219, 384008192000, 414816720885

Offset: 1

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 25 2001

nonn

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

Cf. A064767, A020479, A000252.

f[n_] := (g = First[ Transpose[ FactorInteger[n]]]; n^9*(1 - Apply[ Times, 1 - 1/g] Apply[ Times, 1 - 1/g^2] Apply[ Times, 1 - 1/g^3])); Table[ f[n], {n, 1, 22} ]

a(n) = {my(f = factor(n), p = f[,1], e=f[,2]); n^9 - prod(k = 1, #p, (p[k]-1)*(p[k]^2-1)*(p[k]^3-1)*(p[k]^(9*e[k]-6)));} \\ Amiram Eldar, Aug 03 2024

a(n) = n^9 - A064767(n) = n^9 - n^9 * Product (1-1/p^3)*(1-1/p^2)*(1-1/p) where the product is over all the primes p that divide n.

More terms from Robert G. Wilson v, Nov 30 2001

A070943 Commuting elements: number of ordered pairs g, h in the group GL(2,Z_n) such that gh = hg.

Original entry on oeis.org

1, 18, 384, 1344, 11520, 6912, 96768, 92160, 303264, 207360, 1584000, 516096, 4402944, 1741824, 4423680, 6094848, 22560768, 5458752, 44323200, 15482880, 37158912, 28512000, 141064704, 35389440, 186000000, 79252992, 226748160, 130056192, 572947200, 79626240

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 12 2003

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A000010, A000203, A000252, A062354.

a[n_] := n^4*DivisorSigma[1, n]*EulerPhi[n]*Product[(1-1/p^2)*(1-1/p), {p, FactorInteger[n][[All, 1]]}]; a[1]=1; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, May 02 2013, after Eric M. Schmidt *)

def A070943(n) : return Integer(n^4 * sigma(n) * euler_phi(n) * prod((1-1/p^2)*(1-1/p) for (p,m) in factor(n))) # Eric M. Schmidt, May 02 2013

a(n) = A000252(n) * A062354(n).
a(n) = n^4 * Product_{p prime, p|n} (1-1/p^2)*(1-1/p) * sigma(n)*phi(n).
From Amiram Eldar, Oct 30 2022: (Start)
Multiplicative with a(p^e) = (p^(e+1)-1) * (p-1)^2 * (p+1) * p^(5*e-4).
Sum_{k=1..n} a(k) ~ c * n^7, where c = (1/7) * Product_{p prime} (1 - 1/p^2 - 2/p^3 + 3/p^4 - 1/p^5) = 0.07103214283... . (End)

More terms from Benoit Cloitre, Sep 13 2003

More terms from Eric M. Schmidt, May 02 2013

A085646 Sum of the entries in the character table of the group GL(2,Z_n).

Original entry on oeis.org

1, 5, 24, 52, 120, 120, 336, 496, 654, 600, 1320, 1248, 2184, 1680, 2880

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 11 2003

Cf. A000252, A062354, A053191.

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

A086147 Sum of the orders of the elements in the group GL(2,Z_n).

Original entry on oeis.org

1, 13, 219, 367, 4891, 1977, 36085, 9791, 46731, 39133, 479157, 37119, 1289911, 243703, 375219, 305599, 6991319, 299913, 11500123, 667219, 2610657, 3723423, 40035651, 781127, 14928331, 8544673, 11297307, 4540153, 129539703, 2739477, 209881105, 9748415

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 25 2003

nonn

Eric M. Schmidt, Table of n, a(n) for n = 1..60

Cf. A000252, A036391, A316566.

A086147 := n -> Sum(ConjugacyClasses(GL(2,ZmodnZ(n))), cc->Size(cc) * Order(Representative(cc))); # Eric M. Schmidt, May 18 2013

a(n) = Sum_{k=1..n} k*A316566(n, k). - Andrew Howroyd, Jul 07 2018

Corrected and extended by Eric M. Schmidt, May 18 2013

A229292 Exponent of the group of 2 X 2 invertible matrices over Z/nZ.

Original entry on oeis.org

1, 6, 24, 30, 120, 24, 336, 126, 240, 120, 1320, 120, 2184, 336, 120, 510, 4896, 240, 6840, 120, 336, 1320, 12144, 504, 3120, 2184, 2184, 1680, 24360, 120, 29760, 2046, 1320, 4896, 1680, 240, 50616, 6840, 2184, 2520, 68880, 336, 79464, 1320, 240, 12144

Offset: 1

José María Grau Ribas, Nov 02 2013

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000252, A227477, A316566.

ex[p_, s_] := LCM[p(p^(2 s) - 1), p - 1]; ex[1] := 1; ex[n_] := {aux = 1; Do[aux = LCM[aux, ex[fa[n][[i, 1]], fa[n][[i, 2]]]], {i, 1, Length[fa[n]]}]; aux}[[1]];Table[ex[n], {n, 1, 111}]

a(n)=if(n==1,return(1)); my(f=factor(n)); lcm(vector(#f~,i, f[i,1]*lcm((f[i,1]^(2*f[i,2])-1), f[i,1]-1))) \\ Charles R Greathouse IV, Nov 13 2013

a(p^s) = lcm(p*(p^(2*s) - 1), p - 1); if gcd(m,n)=1 then a(n*m) = lcm(a(n), a(m)).

A327570 a(n) = n*phi(n)^2, phi = A000010.

Original entry on oeis.org

1, 2, 12, 16, 80, 24, 252, 128, 324, 160, 1100, 192, 1872, 504, 960, 1024, 4352, 648, 6156, 1280, 3024, 2200, 11132, 1536, 10000, 3744, 8748, 4032, 22736, 1920, 27900, 8192, 13200, 8704, 20160, 5184, 47952, 12312, 22464, 10240, 65600, 6048, 75852, 17600, 25920, 22264, 99452, 12288

Offset: 1

Jianing Song, Sep 17 2019

a(n) is the order of the group consisting of all upper-triangular (or equivalently, lower-triangular) matrices in GL(2, Z_n). That is to say, a(n) = |G_n|, where G_n = {{{a, b}, {0, d}} : gcd(a, n) = gcd(d, n) = 1}. The group G_n is well-defined because the product of two upper-triangular matrices is again an upper-triangular matrix. For example,{{a, b}, {0, d}} * {{x, y}, {0, z}} = {{a*x, a*y+b*z}, {0, d*z}}.
The exponent of G_n (i.e., the least positive integer k such that x^k = e for all x in G_n) is A174824(n). (Note that {{1, 1}, {0, 1}} is an element with order n and there exists some r such that {{r, 0}, {0, r}} is an element with order psi(n), psi = A002322. It is easy to show that x^lcm(n, psi(n)) = Id = {{1, 0}, {0, 1}} for all x in G_n.)
If only upper-triangular matrices in SL(2, Z_n) are wanted, we get a group of order n*phi(n) = A002618(n) and exponent A174824(n).

			G_3 = {{{1, 0}, {0, 1}}, {{1, 1}, {0, 1}}, {{1, 2}, {0, 1}}, {{1, 0}, {0, 2}}, {{1, 1}, {0, 2}}, {{1, 2}, {0, 2}}, {{2, 0}, {0, 1}}, {{2, 1}, {0, 1}}, {{2, 2}, {0, 1}}, {{2, 0}, {0, 2}}, {{2, 1}, {0, 2}}, {{2, 2}, {0, 2}}} with order 12, so a(3) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A000252, A002618, A174824, A011379, A065464.

Table[n * EulerPhi[n]^2, {n, 1, 100}] (* Amiram Eldar, Sep 19 2020 *)

PARI
```
a(n) = n*eulerphi(n)^2
```

Multiplicative with a(p^e) = (p-1)^2*p^(3e-2).
a(n) = A000010(n)*A002618(n).
a(p) = A011379(p-1) for p prime. - Peter Luschny, Sep 17 2019
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^2/((p-1)^3 * (p^2 + p + 1))) = 1.7394747912949637836019917301710010334604379331855033150372654868327481539... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^4, where c = (1/4) * Product_{p prime} (1 - (2*p-1)/p^3) = A065464 / 4 = 0.1070623764... . - Amiram Eldar, Nov 05 2022

cp's OEIS Frontend

A065501 Number of conjugacy classes in the group SL(2,Z_n) (see A000056).

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A181107 Triangle read by rows: T(n,k) is the number of 2 X 2 matrices over Z(n) having determinant congruent to k mod n, 1 <= n, 0 <= k <= n-1.

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PARI

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A327568 Exponent of the group GL(2, Z_n).

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A020479 Number of noninvertible 2 X 2 matrices over Z/nZ (determinant is a divisor of 0).

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Mathematica

PARI

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A065499 Noninvertible 3 X 3 matrices over Z_n.

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PARI

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A070943 Commuting elements: number of ordered pairs g, h in the group GL(2,Z_n) such that gh = hg.

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Mathematica

Sage

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A085646 Sum of the entries in the character table of the group GL(2,Z_n).

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A086147 Sum of the orders of the elements in the group GL(2,Z_n).

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GAP