A064767 -id:A064767 - cp's OEIS frontend

A065499 Noninvertible 3 X 3 matrices over Z_n.

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

A115226 Order of the group of invertible 3 X 3 symmetric matrices over Z(n).

Original entry on oeis.org

1, 28, 468, 1792, 12400, 13104, 100548, 114688, 341172, 347200, 1609300, 838656, 4453488, 2815344, 5803200, 7340032, 22713088, 9552816, 44563284, 22220800, 47056464, 45060400, 141587908, 53673984, 193750000, 124697664, 248714388, 180182016, 574288624, 162489600

Offset: 1

T. D. Noe, Jan 16 2006

Note that A115225 gives the number of 3 x 3 symmetric matrices having nonzero determinant. However, for composite n, a nonzero determinant is not sufficient for the matrix to be invertible; the determinant must also be relatively prime to n.

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

Cf. A000056 (order of the group SL(2, Z_n)), A064767 (order of the group GL(3, Z_n)), A115225.

Table[cnt=0; Do[m={{a, b, c}, {b, d, e}, {c, e, f}}; If[Det[m, Modulus->n]>0 && MatrixQ[Inverse[m, Modulus->n]], 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}]
f[p_, e_] := p^(6*e - 4)*(p^3 - 1)*(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 10 2020 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(6*f[i,2] - 4)*(f[i,1]^3 - 1)*(f[i,1] - 1));} \\ Amiram Eldar, Nov 05 2022

For prime p, a(p) = (p^3-1)*(p-1)*p^2.
In general, a(n) = A115224(n) * phi(n) = A064767(n)/A000056(n).
Multiplicative with a(p^e) = p^(6*e - 4)*(p^3 - 1)*(p - 1). - Amiram Eldar, Sep 10 2020
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^4/((p-1)^3 * (p^2+p+1)^2 * (p^3+1))) = 1.03859354030263389220782701124174403591851545785245128014455467710993780757... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^7, where c = (1/7) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.08230753362... . - Amiram Eldar, Nov 05 2022

More terms from Amiram Eldar, Sep 10 2020

A138565 Array read by rows: T(n,k) is the number of automorphisms of the k-th Abelian 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, 4, 8, 168, 6, 48, 4, 10, 4, 12, 12, 6, 8, 8, 16, 96, 192, 20160, 16, 6, 48, 18, 8, 24, 12, 10, 22, 8, 16, 336, 20, 480, 12, 18, 108, 11232, 12, 36, 28, 8, 30, 16, 32, 128, 384, 1536, 21504, 9999360, 20, 16, 24, 12, 36, 96, 288, 36, 18, 24, 16, 32, 672

Offset: 1

Benoit Jubin, May 12 2008

This is a subtable of A137316.
The length of the n-th row is A000688(n).
The largest value of the n-th row is A061350(n).
The number phi(n) = A000010(n) appears in the n-th row.
The number A064767(n) appears in the (n^3)-th row.
The number A062771(n) appears in the (2n)-th row.

			The table begins as follows:
1
1
2
2 6
4
2
6
4 8 168
6 48
4
10
4 12
The first row with two numbers corresponds to the two Abelian groups of order 4, the cyclic group C_4 and the Klein group C_2 x C_2, whose automorphism groups are respectively the group (C_4)^x = C_2 and the symmetric group S_3.

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

Print("\n") ;
for o in [ 1 .. 40 ] do
    n := NumberSmallGroups(o) ;
    og := [] ;
    for i in [1 .. n] do
        g := SmallGroup(o,i) ;
        if IsAbelian(g) then
            H := AutomorphismGroup(g) ;
            ho := Order(H) ;
            Add(og,ho) ;
        fi ;
    od;
    Sort(og) ;
    Print(og) ;
    Print("\n") ;
od; # R. J. Mathar, Jul 13 2013

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A065499 Noninvertible 3 X 3 matrices over Z_n.

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A115226 Order of the group of invertible 3 X 3 symmetric matrices over Z(n).

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A138565 Array read by rows: T(n,k) is the number of automorphisms of the k-th Abelian group of order n, where the ordering is such that the rows are nondecreasing.

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