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A115224 Number of 3 X 3 symmetric matrices over Z(n) having determinant 1. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Enrique Pérez Herrero, Table of n, a(n) for n = 1..4885

FORMULA

a(1)=1 because the matrix of all zeros has determinant 0, but 0=1 (mod 1). For prime n, a(n)=(n^3-1)n^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

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 *)

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

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Last modified June 16 04:54 EDT 2019. Contains 324145 sequences. (Running on oeis4.)