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

1,2

Links

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

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

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 *)

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)).

Sequence in context: A042524 A125365 A126523 * A135497 A138405 A024015

Adjacent sequences: A115221 A115222 A115223 * A115225 A115226 A115227

Keyword

mult,nonn

Author

T. D. Noe, Jan 16 2006

Status

approved