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A364771
Order of the symplectic group of 4 X 4 matrices over Z_n.
1
1, 720, 51840, 737280, 9360000, 37324800, 276595200, 754974720, 3061100160, 6739200000, 25721308800, 38220595200, 137037962880, 199148544000, 485222400000, 773094113280, 2008994088960, 2203992115200, 6114035779200, 6900940800000, 14338695168000, 18519342336000, 41348052472320
OFFSET
1,2
COMMENTS
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.
LINKS
E. Artin, Geometric Algebra, Wiley Classics Library. John Wiley & Sons, Inc., New York, 1988. Reprint of the 1957 original, A Wiley-Interscience Publication.
Larry C. Grove, Classical Groups and Geometric Algebra, Grad. Stud. Math., 39 American Mathematical Society, Providence, RI, 2002. x+169 pp.
FORMULA
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.
For primes p : a(p) = A003931(n), where A246655(n) = p.
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
MATHEMATICA
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 *)
PROG
(Sage)
def a(n):
return product([p^(10*n.valuation(p)-6)*(p^2 - 1)*(p^4 - 1)
for p in n.prime_factors()])
CROSSREFS
Sequence in context: A246614 A291113 A246217 * A003931 A158049 A294705
KEYWORD
nonn,mult
AUTHOR
Robin Visser, Aug 06 2023
STATUS
approved