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A115226 Order of the group of invertible 3 X 3 symmetric matrices over Z(n). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

LINKS

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

FORMULA

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

MATHEMATICA

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 *)

CROSSREFS

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

Sequence in context: A079518 A320820 A160060 * A086782 A115225 A223997

Adjacent sequences:  A115223 A115224 A115225 * A115227 A115228 A115229

KEYWORD

mult,nonn,easy

AUTHOR

T. D. Noe, Jan 16 2006

EXTENSIONS

More terms from Amiram Eldar, Sep 10 2020

STATUS

approved

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Last modified March 3 04:26 EST 2021. Contains 341756 sequences. (Running on oeis4.)