A206719 - OEIS

%I #12 Dec 18 2017 11:56:43

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

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

%H Antti Karttunen, <a href="/A206719/b206719.txt">Table of n, a(n) for n = 1..65537</a>

%e p(1,n) = 1, so a(1)=0

%e p(2,n) = x, so a(2)=1

%e p(6,n) = x(1+x), so a(6)=2

%e p(18,n) = x(x+1)(1-x+x^2), so a(18)=3

%e p(90,n) = x(1+x)(1+x^2)(1-x+x^2), so a(90)=4

%t t = Table[IntegerDigits[n, 2], {n, 1, 1000}];

%t b[n_] := Reverse[Table[x^k, {k, 0, n}]]

%t p[n_, x_] := p[n, x] = t[[n]].b[-1 + Length[t[[n]]]]

%t TableForm[Table[{n, p[n, x],

%t    FactorList[p[n, x]], -1 + Length[FactorList[p[n, x]]]}, {n, 1, 9}]]

%t Table[Length[FactorList[p[n, x]]], {n, 1, 120}]

%o (PARI) A206719(n) = { my(f = factor(Pol(binary(n)))); (#f~); }; \\ _Antti Karttunen_, Dec 16 2017

%Y Cf. A206073, A206074, A257000.

%Y Cf. also A091221, A206442.

%K nonn

%O 1,6

%A _Clark Kimberling_, Feb 11 2012