A011786 - OEIS

%I #32 Oct 23 2022 02:54:25

%S 1,20160,12130560,660602880,29016000000,244552089600,4635182361600,

%T 21646635171840,174060277297920,584962560000000,4139330225184000,

%U 8013482872012800,50858076935877120,93445276409856000,351980328960000000,709316941310853120,2851903720876769280

%N Number of 4 X 4 matrices whose determinant is 1 mod n.

%C Order of the group SL(4,Z_n). For n > 2, a(n) is divisible by 11520. - _Jianing Song_, Nov 24 2018

%H T. D. Noe, <a href="/A011786/b011786.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = (n^16/phi(n))*Product_{primes p dividing n} ((1 - 1/p^4)*(1 - 1/p^3)*(1 - 1/p^2)*(1 - 1/p)). Multiplicative with a(p^e) = p^(15*e-9)*(p^4 - 1)*(p^3 - 1)*(p^2 - 1). - _Vladeta Jovovic_, Nov 18 2001

%F a(n) = n^15*Product_{primes p dividing n} ((1 - 1/p^4)*(1 - 1/p^3)*(1 - 1/p^2)) = A305186(n)/phi(n). - _Jianing Song_, Nov 24 2018

%F Sum_{k=1..n} a(k) ~ c * n^16, where c = (1/16) * Product_{p prime} ((p^10 - p^7 - p^6 - p^5 + p^4 + p^3 + p^2 - 1)/p^10) = 0.04715136234... . - _Amiram Eldar_, Oct 23 2022

%t f[p_, e_] := (1 - 1/p^4)*(1 - 1/p^3)*(1 - 1/p^2); a[1] = 1; a[n_] := n^15 * Times @@ f @@@ FactorInteger[n]; Array[a, 17] (* _Amiram Eldar_, Oct 23 2022 *)

%o (PARI) a(n) = f = factor(n); n^16/eulerphi(n) * prod(i=1, #f~, (1-1/f[i,1]^4)*(1-1/f[i,1]^3)*(1-1/f[i,1]^2)*(1-1/f[i,1])); \\ _Michel Marcus_, Sep 02 2013

%Y Cf. A000056 (SL(2,Z_n)), A011785 (SL(3,Z_n)).

%Y Cf. A000252 (GL(2,Z_n)), A064767 (GL(3,Z_n)), A305186 (GL(4,Z_n)).

%Y Cf. A000010.

%K nonn,mult

%O 1,2

%A benlove(AT)preston.polaristel.net (Benjamin T. Love)

%E More terms from _Vladeta Jovovic_, Nov 18 2001