cp's OEIS Frontend

A294705 Order of the general symplectic group of 4 X 4 matrices over Z_n.

%I A294705 #33 Aug 07 2023 02:11:41
%S A294705 1,720,103680,1474560,37440000,74649600,1659571200,3019898880,
%T A294705 18366600960,26956800000,257213088000,152882380800,1644455554560,
%U A294705 1194891264000,3881779200000,6184752906240,32143905423360,13223952691200,110052644025600,55207526400000,172064342016000,185193423360000
%N A294705 Order of the general symplectic group of 4 X 4 matrices over Z_n.
%C A294705 Let M be any fixed nonsingular skew-symmetric 4 X 4 matrix over the integers mod n. Then a(n) is the number of invertible 4 X 4 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 A294705 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 A294705 a(n) = Product_{primes p dividing n} p^(11*v_p(n) - 7)*(p - 1)*(p^2 - 1)*(p^4 - 1), where v_p(n) is the largest power k such that p^k divides n.
%F A294705 Sum_{k=1..n} a(k) ~ c * n^12 / 12, where c = Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4 - 1/p^5 + 1/p^6 + 1/p^7 - 1/p^8) = 0.5251079212... . - _Amiram Eldar_, Aug 07 2023
%t A294705 f[p_, e_] := p^(11*e - 7)*(p - 1)*(p^2 - 1)*(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* _Amiram Eldar_, Aug 07 2023 *)
%o A294705 (Sage)
%o A294705 def a(n):
%o A294705     return product([p^(11*n.valuation(p) - 7)*(p - 1)*(p^2 - 1)*(p^4 - 1)
%o A294705         for p in n.prime_factors()])
%Y A294705 Cf. A305186.
%K A294705 nonn,mult
%O A294705 1,2
%A A294705 _Robin Visser_, Aug 06 2023