A206829 - OEIS

%I #7 Mar 30 2012 18:58:12

%S 0,1,1,1,2,2,1,1,2,1,1,2,3,1,2,1,2,1,2,2,1,2,1,2,2,1,3,3,1,3,1,2,2,2,

%T 1,2,2,2,2,2,1,1,3,1,2,2,2,1,2,1,2,2,2,1,3,1,1,2,4,1,3,1,2,2,3,1,2,2,

%U 2,1,3,1,2,2,2,1,2,1,3,1,2,1,3,1,2,1,2,1,2,2,2,3,1,2,2,2,1,3,1

%N Number of distinct irreducible factors of the polynomial y(n,x) defined at A206821.

%C The first 6 polynomials: 1, x, 1+x, x^2, x^2-1, x^2-x, representing an ordering of the monic polynomials having coefficients in {-1,0,1}; see A206821.

%e y(5,x) = (x-1)(x+1), so a(5)=2.

%t t = Table[IntegerDigits[n, 2], {n, 1, 1000}];

%t b[n_] := Reverse[Table[x^k, {k, 0, n}]]

%t p[n_] := p[n] = t[[n]].b[-1 + Length[t[[n]]]]

%t TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]]

%t f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1];

%t q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]]

%t q2[n_] := p[n - f[k] + 2]

%t y1 = Table[p[n], {n, 1, 4}];

%t Do[AppendTo[y1, Join[Table[q1[n], {n, f[k], g[k] - 1}],

%t    Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 8}]

%t y = Flatten[y1]; (* polynomials over {-1,0,1} *)

%t TableForm[Table[{n, y[[n]], Factor[y[[n]]]}, {n, 1, 10}]]

%t Table[-1 + Length[FactorList[y[[n]]]],

%t {n, 1, 120}]  (* A206829 *)

%Y Cf. A206821.

%K nonn

%O 1,5

%A _Clark Kimberling_, Feb 12 2012