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

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

Author

T. D. Noe, Jan 16 2006

Keywords

• mult
• nonn

Links

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

Crossrefs

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)).

Programs

Mathematica

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 *)

Formula

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