cp's OEIS Frontend

A206829 Number of distinct irreducible factors of the polynomial y(n,x) defined at A206821.

%I A206829 #7 Mar 30 2012 18:58:12
%S A206829 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 A206829 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 A206829 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 A206829 Number of distinct irreducible factors of the polynomial y(n,x) defined at A206821.
%C A206829 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 A206829 y(5,x) = (x-1)(x+1), so a(5)=2.
%t A206829 t = Table[IntegerDigits[n, 2], {n, 1, 1000}];
%t A206829 b[n_] := Reverse[Table[x^k, {k, 0, n}]]
%t A206829 p[n_] := p[n] = t[[n]].b[-1 + Length[t[[n]]]]
%t A206829 TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]]
%t A206829 f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1];
%t A206829 q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]]
%t A206829 q2[n_] := p[n - f[k] + 2]
%t A206829 y1 = Table[p[n], {n, 1, 4}];
%t A206829 Do[AppendTo[y1, Join[Table[q1[n], {n, f[k], g[k] - 1}],
%t A206829    Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 8}]
%t A206829 y = Flatten[y1]; (* polynomials over {-1,0,1} *)
%t A206829 TableForm[Table[{n, y[[n]], Factor[y[[n]]]}, {n, 1, 10}]]
%t A206829 Table[-1 + Length[FactorList[y[[n]]]],
%t A206829 {n, 1, 120}]  (* A206829 *)
%Y A206829 Cf. A206821.
%K A206829 nonn
%O A206829 1,5
%A A206829 _Clark Kimberling_, Feb 12 2012