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%I A020478 #40 Oct 22 2020 02:52:44
%S A020478 1,10,33,88,145,330,385,736,945,1450,1441,2904,2353,3850,4785,6016,
%T A020478 5185,9450,7201,12760,12705,14410,12673,24288,18625,23530,26001,33880,
%U A020478 25201,47850,30721,48640,47553,51850,55825,83160,51985,72010,77649,106720
%N A020478 Number of singular 2 X 2 matrices over Z(n) (i.e., with determinant = 0).
%H A020478 Amiram Eldar, <a href="/A020478/b020478.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%F A020478 From _Vladeta Jovovic_, Apr 22 2002: (Start)
%F A020478 a(n) = n^4 - A005353(n).
%F A020478 Multiplicative with a(p^e) = p^(2*e - 1)*(p^(e+1) + p^e - 1). (End)
%F A020478 Dirichlet g.f.: zeta(s-2)*zeta(s-3)/zeta(s-1).
%F A020478 A102631(n) | a(n). - _R. J. Mathar_, Mar 30 2011
%F A020478 Sum_{k=1..n} a(k) ~ Pi^2 * n^4 / (24*Zeta(3)). - _Vaclav Kotesovec_, Jan 31 2019
%F A020478 From _Piotr Rysinski_, Sep 11 2020: (Start)
%F A020478 a(n) = n * A069097(n).
%F A020478 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)
%t A020478 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 *)
%o A020478 (PARI) a(n)=if(n<1, 0, direuler(p=2, n, (1-p*X)/((1-p^2*X)*(1-p^3*X)))[n])
%o A020478 (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
%Y A020478 Cf. A005353, A059306, A062801, A069097, A102631, A240547.
%K A020478 nonn,mult,easy
%O A020478 1,2
%A A020478 _David W. Wilson_

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