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

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, 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, 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

Offset

1,5

Comments

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.

Links

Table of n, a(n) for n=1..99.

Example

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

Mathematica

t = Table[IntegerDigits[n, 2], {n, 1, 1000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_] := p[n] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]]
f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1];
q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]]
q2[n_] := p[n - f[k] + 2]
y1 = Table[p[n], {n, 1, 4}];
Do[AppendTo[y1, Join[Table[q1[n], {n, f[k], g[k] - 1}],
   Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 8}]
y = Flatten[y1]; (* polynomials over {-1, 0, 1} *)
TableForm[Table[{n, y[[n]], Factor[y[[n]]]}, {n, 1, 10}]]
Table[-1 + Length[FactorList[y[[n]]]],
{n, 1, 120}]  (* A206829 *)

Crossrefs

Cf. A206821.

Sequence in context: A336123 A353849 A075661 * A319694 A335641 A163495

Adjacent sequences: A206826 A206827 A206828 * A206830 A206831 A206832

Keyword

nonn

Author

Clark Kimberling, Feb 12 2012

Status

approved