%I #14 Aug 03 2024 07:10:03

%S 10,33,160,145,1008,385,2560,2673,7120,1441,16128,2353,26320,27585,

%T 40960,5185,81648,7201,113920,97713,155056,12673,258048,90625,299728,

%U 216513,421120,25201,671760,30721,655360,552321,866320,532945,1306368,51985

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

%F 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]

%F a(n) = n^4 - A000252(n). - _T. D. Noe_, Jan 16 2006

%t 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 *)

%o (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

%Y Cf. A000252.

%K nonn

%O 2,1

%A _David W. Wilson_