A364782 - OEIS

%I #11 Aug 08 2023 03:22:25

%S 1,1451520,18341406720,6088116142080,1828008000000000,

%T 26622918682214400,3281486623259443200,25535409887190712320,

%U 575572777593233172480,2653390172160000000000,73385854415869121280000,111664614320486586777600,2947127504061746732912640,4763143463393546993664000

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

%C 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.

%H Hanson Smith, <a href="https://doi.org/10.1142/S1793042122501172">Frobenius finds non-monogenic division fields of abelian varieties</a>, Int. J. Number Theory 18 (2022), no. 10, 2299-2315.

%F 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.

%F 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

%t 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 *)

%o (Sage)

%o def a(n):

%o     return product([p^(22*n.valuation(p)-13)*(p-1)*(p^2-1)*(p^4-1)*(p^6-1)

%o         for p in n.prime_factors()])

%Y Cf. A294705, A364783.

%K nonn,mult

%O 1,2

%A _Robin Visser_, Aug 07 2023