A020479 Number of noninvertible 2 X 2 matrices over Z/nZ (determinant is a divisor of 0).

10, 33, 160, 145, 1008, 385, 2560, 2673, 7120, 1441, 16128, 2353, 26320, 27585, 40960, 5185, 81648, 7201, 113920, 97713, 155056, 12673, 258048, 90625, 299728, 216513, 421120, 25201, 671760, 30721, 655360, 552321, 866320, 532945, 1306368, 51985

Offset: 2

Author

David W. Wilson

Keywords

• nonn

Crossrefs

Cf. A000252.

Programs

Mathematica

f[p_, e_] := (p - 1)^2*(p + 1)*p^(4*e - 3); a[n_] := n^4 - Times @@ f @@@ FactorInteger[n]; Array[a, 36, 2] (* Amiram Eldar, Aug 03 2024 *)

PARI

a(n) = {my(f = factor(n), p = f[,1], e=f[,2]); n^4 - prod(k = 1, #p, (p[k] - 1)^2*(p[k] + 1)*p[k]^(4*e[k] - 3));} \\ Amiram Eldar, Aug 03 2024

Formula

a(n) seems to be divisible by n. - Ralf Stephan, Sep 01 2003 [This is true and can be easily proven from the formula below and from the multiplicative formula for A000252(n). - Amiram Eldar, Aug 03 2024]

a(n) = n^4 - A000252(n). - T. D. Noe, Jan 16 2006