A020478 Number of singular 2 X 2 matrices over Z(n) (i.e., with determinant = 0).

1, 10, 33, 88, 145, 330, 385, 736, 945, 1450, 1441, 2904, 2353, 3850, 4785, 6016, 5185, 9450, 7201, 12760, 12705, 14410, 12673, 24288, 18625, 23530, 26001, 33880, 25201, 47850, 30721, 48640, 47553, 51850, 55825, 83160, 51985, 72010, 77649, 106720

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• mult
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Crossrefs

Cf. A005353, A059306, A062801, A069097, A102631, A240547.

Programs

Mathematica

f[p_, e_] := p^(2*e - 1)*(p^(e + 1) + p^e - 1); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2020 *)

PARI

a(n)=if(n<1, 0, direuler(p=2, n, (1-p*X)/((1-p^2*X)*(1-p^3*X)))[n])

PARI

a(n)=local(c=0); forvec(x=vector(4,k,[1,n]),c+=((x[1]*x[2]-x[3]*x[4])%n==0)); c

Formula

From Vladeta Jovovic, Apr 22 2002: (Start)

a(n) = n^4 - A005353(n).

Multiplicative with a(p^e) = p^(2*e - 1)*(p^(e+1) + p^e - 1). (End)

Dirichlet g.f.: zeta(s-2)*zeta(s-3)/zeta(s-1).

A102631(n) | a(n). - R. J. Mathar, Mar 30 2011

Sum_{k=1..n} a(k) ~ Pi^2 * n^4 / (24*Zeta(3)). - Vaclav Kotesovec, Jan 31 2019

From Piotr Rysinski, Sep 11 2020: (Start)

a(n) = n * A069097(n).

Proof: a(n) is multiplicative with a(p^e) = p^(2*e - 1)*(p^(e+1) + p^e - 1), A069097(n) is multiplicative with A069097(p^e) = p^(e-1)*(p^e*(p+1)-1), so a(p^e) = p^e*A069097(p^e). (End)