A206829 - OEIS

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

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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]