A364782 Order of the general symplectic group of 6 X 6 matrices over Z_n.

1, 1451520, 18341406720, 6088116142080, 1828008000000000, 26622918682214400, 3281486623259443200, 25535409887190712320, 575572777593233172480, 2653390172160000000000, 73385854415869121280000, 111664614320486586777600, 2947127504061746732912640, 4763143463393546993664000

Offset

1,2

Comments

Let M be any fixed nonsingular skew-symmetric 6 X 6 matrix over the integers mod n. Then a(n) is the number of invertible 6 X 6 matrices A over the integers mod n such that A^T * M * A = c*M for some nonzero constant c (mod n), where A^T denotes the transpose of A.

Links

Table of n, a(n) for n=1..14.

Hanson Smith, Frobenius finds non-monogenic division fields of abelian varieties, Int. J. Number Theory 18 (2022), no. 10, 2299-2315.

Formula

a(n) = Product_{primes p dividing n} p^(22*v_p(n) - 13)*(p - 1)*(p^2 - 1)*(p^4 - 1)*(p^6 - 1), where v_p(n) is the largest power k such that p^k divides n.

Sum_{k=1..n} a(k) ~ c * n^23 / 23, where c = Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4 - 1/p^5 + 1/p^6 + 1/p^9 - 1/p^10 + 1/p^11 - 1/p^12 - 1/p^13 + 1/p^14) = 0.5228053524... . - Amiram Eldar, Aug 08 2023

Mathematica

f[p_, e_] := p^(22*e - 13)*(p - 1)*(p^2 - 1)*(p^4 - 1)*(p^6 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 15] (* Amiram Eldar, Aug 08 2023 *)

Prog

(Sage)
def a(n):
    return product([p^(22*n.valuation(p)-13)*(p-1)*(p^2-1)*(p^4-1)*(p^6-1)
        for p in n.prime_factors()])

Crossrefs

Cf. A294705, A364783.

Sequence in context: A003939 A364783 A003932 * A205044 A237314 A263071

Adjacent sequences: A364779 A364780 A364781 * A364783 A364784 A364785

Keyword

nonn,mult

Author

Robin Visser, Aug 07 2023

Status

approved