A011786 Number of 4 X 4 matrices whose determinant is 1 mod n.

1, 20160, 12130560, 660602880, 29016000000, 244552089600, 4635182361600, 21646635171840, 174060277297920, 584962560000000, 4139330225184000, 8013482872012800, 50858076935877120, 93445276409856000, 351980328960000000, 709316941310853120, 2851903720876769280

Offset: 1

Author

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

Keywords

• nonn
• mult

Comments

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

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

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

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

Cf. A000010.

Programs

Mathematica

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 *)

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

Python

from math import prod
from sympy import factorint
def A011786(n): return prod(p**(3*(5*e-3))*(p**2*(p*(p*(p*(p*(p*(p-1)*(p+1)-1)-1)+1)+1)+1)-1) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

Formula

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

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

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

Extensions

More terms from Vladeta Jovovic, Nov 18 2001