A005353 Number of 2 X 2 matrices with entries mod n and nonzero determinant. (Formerly M4254)

0, 6, 48, 168, 480, 966, 2016, 3360, 5616, 8550, 13200, 17832, 26208, 34566, 45840, 59520, 78336, 95526, 123120, 147240, 181776, 219846, 267168, 307488, 372000, 433446, 505440, 580776, 682080, 762150, 892800, 999936, 1138368, 1284486

Offset

1,2

References

T. Brenner, personal communication.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

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

Richard K. Guy, Letter to N. J. A. Sloane, Sep 1989.

Formula

a(n) = n^4 - A020478(n).

For prime n, a(n) = (n^2-1)(n-1)n. - T. D. Noe, Jan 12 2006

Mathematica

Table[cnt=0; Do[m={{a, b}, {c, d}}; If[Det[m, Modulus->p] > 0, cnt++ ], {a, 0, p-1}, {b, 0, p-1}, {c, 0, p-1}, {d, 0, p-1}]; cnt, {p, 37}] (* T. D. Noe, Jan 12 2006 *)
f[p_, e_] := p^(2*e - 1)*(p^(e + 1) + p^e - 1); a[1] = 0; a[n_] := n^4 - Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 31 2023 *)

Prog

(PARI) a(n) = {my(f = factor(n), p, e); n^4 - prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2];  p^(2*e - 1)*(p^(e + 1) + p^e - 1)); } \\ Amiram Eldar, Oct 31 2023

Crossrefs

Cf. A020478, A059306, A062801.

Sequence in context: A244726 A335384 A331668 * A047927 A059238 A371067

Adjacent sequences: A005350 A005351 A005352 * A005354 A005355 A005356

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, R. K. Guy

Extensions

More terms from T. D. Noe, Jan 12 2006

Status

approved