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A020478
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Number of singular 2 X 2 matrices over Z(n) (i.e., with determinant = 0).
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6
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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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).
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)
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,mult,easy
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AUTHOR
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STATUS
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approved
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