A206719 Number of distinct irreducible factors of the polynomial p(n,x) defined at A206073.

0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 3, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 3, 1, 2, 3, 1, 2, 1, 3, 1, 3, 1, 2, 3, 2, 1, 3, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 4, 1, 2, 2, 2, 2, 2, 1, 3, 2

Offset: 1

Author

Clark Kimberling, Feb 11 2012

Keywords

• nonn

Comments

The polynomials having coefficients in {0,1} are enumerated as in A206074 (and A206073).

Examples

			p(1,n) = 1, so a(1)=0
p(2,n) = x, so a(2)=1
p(6,n) = x(1+x), so a(6)=2
p(18,n) = x(x+1)(1-x+x^2), so a(18)=3
p(90,n) = x(1+x)(1+x^2)(1-x+x^2), so a(90)=4
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A206073, A206074, A257000.

Cf. also A091221, A206442.

Programs

Mathematica

t = Table[IntegerDigits[n, 2], {n, 1, 1000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := p[n, x] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n, x],
   FactorList[p[n, x]], -1 + Length[FactorList[p[n, x]]]}, {n, 1, 9}]]
Table[Length[FactorList[p[n, x]]], {n, 1, 120}]

PARI

A206719(n) = { my(f = factor(Pol(binary(n)))); (#f~); }; \\ Antti Karttunen, Dec 16 2017