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A364771 Order of the symplectic group of 4 X 4 matrices over Z_n.

%I A364771 #13 Aug 07 2023 02:12:37
%S A364771 1,720,51840,737280,9360000,37324800,276595200,754974720,3061100160,
%T A364771 6739200000,25721308800,38220595200,137037962880,199148544000,
%U A364771 485222400000,773094113280,2008994088960,2203992115200,6114035779200,6900940800000,14338695168000,18519342336000,41348052472320
%N A364771 Order of the symplectic group of 4 X 4 matrices over Z_n.
%C A364771 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 = M, where A^T denotes the transpose of A.
%H A364771 E. Artin, <a href="https://doi.org/10.1002/9781118164518">Geometric Algebra</a>, Wiley Classics Library. John Wiley & Sons, Inc., New York, 1988. Reprint of the 1957 original, A Wiley-Interscience Publication.
%H A364771 Larry C. Grove, <a href="https://doi.org/10.1090/gsm/039">Classical Groups and Geometric Algebra</a>, Grad. Stud. Math., 39 American Mathematical Society, Providence, RI, 2002. x+169 pp.
%F A364771 a(n) = Product_{primes p dividing n} p^(10*v_p(n) - 6)*(p^2 - 1)*(p^4 - 1), where v_p(n) is the largest power k such that p^k divides n.
%F A364771 For primes p : a(p) = A003931(n), where A246655(n) = p.
%F A364771 Sum_{k=1..n} a(k) ~ c * n^11 / 11, where c = Product_{p prime} (1 - (p^4+p^2-1)/p^7) = 0.8062434963... . - _Amiram Eldar_, Aug 07 2023
%t A364771 f[p_, e_] := p^(10*e - 6)*(p^2 - 1)*(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* _Amiram Eldar_, Aug 07 2023 *)
%o A364771 (Sage)
%o A364771 def a(n):
%o A364771     return product([p^(10*n.valuation(p)-6)*(p^2 - 1)*(p^4 - 1)
%o A364771         for p in n.prime_factors()])
%Y A364771 Cf. A011786, A305186.
%K A364771 nonn,mult
%O A364771 1,2
%A A364771 _Robin Visser_, Aug 06 2023