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

1,6

Comments

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

Links

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

Example

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

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}]

Prog

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

Crossrefs

Cf. A206073, A206074, A257000.

Cf. also A091221, A206442.

Sequence in context: A325169 A322862 A325225 * A240086 A322306 A359778

Adjacent sequences: A206716 A206717 A206718 * A206720 A206721 A206722

Keyword

nonn

Author

Clark Kimberling, Feb 11 2012

Status

approved