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%I #26 Nov 05 2022 08:18:54
%S 1,28,468,1792,12400,13104,100548,114688,341172,347200,1609300,838656,
%T 4453488,2815344,5803200,7340032,22713088,9552816,44563284,22220800,
%U 47056464,45060400,141587908,53673984,193750000,124697664,248714388,180182016,574288624,162489600
%N Order of the group of invertible 3 X 3 symmetric matrices over Z(n).
%C 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.
%H Amiram Eldar, <a href="/A115226/b115226.txt">Table of n, a(n) for n = 1..10000</a>
%F For prime p, a(p) = (p^3-1)*(p-1)*p^2.
%F In general, a(n) = A115224(n) * phi(n) = A064767(n)/A000056(n).
%F Multiplicative with a(p^e) = p^(6*e - 4)*(p^3 - 1)*(p - 1). - _Amiram Eldar_, Sep 10 2020
%F 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
%F Sum_{k=1..n} a(k) ~ c * n^7, where c = (1/7) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.08230753362... . - _Amiram Eldar_, Nov 05 2022
%t 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}]
%t 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 *)
%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(6*f[i,2] - 4)*(f[i,1]^3 - 1)*(f[i,1] - 1));} \\ _Amiram Eldar_, Nov 05 2022
%Y Cf. A000056 (order of the group SL(2, Z_n)), A064767 (order of the group GL(3, Z_n)), A115225.
%K mult,nonn,easy
%O 1,2
%A _T. D. Noe_, Jan 16 2006
%E More terms from _Amiram Eldar_, Sep 10 2020
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