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%I #12 Aug 08 2023 03:22:29
%S 1,1451520,9170703360,3044058071040,457002000000000,13311459341107200,
%T 546914437209907200,6383852471797678080,95928796265538862080,
%U 663347543040000000000,7338585441586912128000,27916153580121646694400,245593958671812227742720,793857243898924498944000
%N Order of the 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 = M, where A^T denotes the transpose of A.
%H 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 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 a(n) = Product_{primes p dividing n} p^(21*v_p(n) - 12)*(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 For primes p : a(p) = A003932(n), where A246655(n) = p.
%F Sum_{k=1..n} a(k) ~ c * n^22 / 22, where c = Product_{p prime} (1 - 1/p^3 - 1/p^5 + 1/p^9 + 1/p^11 - 1/p^13) = 0.8006965549... . - _Amiram Eldar_, Aug 08 2023
%t f[p_, e_] := p^(21*e - 12)*(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^(21*n.valuation(p) - 12)*(p^2 - 1)*(p^4 - 1)*(p^6 - 1)
%o for p in n.prime_factors()])
%Y Cf. A364771, A364782.
%K nonn,mult
%O 1,2
%A _Robin Visser_, Aug 07 2023
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