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 A020478 Number of singular 2 X 2 matrices over Z(n) (i.e., with determinant = 0). 6
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe) 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) 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 *) PROG (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 CROSSREFS Cf. A005353, A059306, A062801, A069097, A102631, A240547. Sequence in context: A367014 A162433 A003012 * A094170 A373129 A004638 Adjacent sequences: A020475 A020476 A020477 * A020479 A020480 A020481 KEYWORD nonn,mult,easy AUTHOR David W. Wilson STATUS approved

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Last modified August 2 22:39 EDT 2024. Contains 374875 sequences. (Running on oeis4.)