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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, 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
KEYWORD
mult,nonn
AUTHOR
T. D. Noe, Jan 16 2006
STATUS
approved