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

%I A206719 #12 Dec 18 2017 11:56:43
%S A206719 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,
%T A206719 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,
%U A206719 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
%N A206719 Number of distinct irreducible factors of the polynomial p(n,x) defined at A206073.
%C A206719 The polynomials having coefficients in {0,1} are enumerated as in A206074 (and A206073).
%H A206719 Antti Karttunen, <a href="/A206719/b206719.txt">Table of n, a(n) for n = 1..65537</a>
%e A206719 p(1,n) = 1, so a(1)=0
%e A206719 p(2,n) = x, so a(2)=1
%e A206719 p(6,n) = x(1+x), so a(6)=2
%e A206719 p(18,n) = x(x+1)(1-x+x^2), so a(18)=3
%e A206719 p(90,n) = x(1+x)(1+x^2)(1-x+x^2), so a(90)=4
%t A206719 t = Table[IntegerDigits[n, 2], {n, 1, 1000}];
%t A206719 b[n_] := Reverse[Table[x^k, {k, 0, n}]]
%t A206719 p[n_, x_] := p[n, x] = t[[n]].b[-1 + Length[t[[n]]]]
%t A206719 TableForm[Table[{n, p[n, x],
%t A206719    FactorList[p[n, x]], -1 + Length[FactorList[p[n, x]]]}, {n, 1, 9}]]
%t A206719 Table[Length[FactorList[p[n, x]]], {n, 1, 120}]
%o A206719 (PARI) A206719(n) = { my(f = factor(Pol(binary(n)))); (#f~); }; \\ _Antti Karttunen_, Dec 16 2017
%Y A206719 Cf. A206073, A206074, A257000.
%Y A206719 Cf. also A091221, A206442.
%K A206719 nonn
%O A206719 1,6
%A A206719 _Clark Kimberling_, Feb 11 2012