A107751

[Robert Israel]

You may have noticed that (at least for n up to 200) whenever your 
polynomial x^n+x+1 factors, one of the factors is x^2+x+1.  I don't
know if this is true for all n, but if so there's doubtless a good
reason for it.  In any case it's easy to see that for n >= 2, x^n+x+1
is divisible by x^2+x+1 if and only if n == 2 mod 3.

Robert Israel                                israel at math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada

On Sat, 13 Oct 2007, Artur wrote:

> I would like to ask is somebody which understand A107751 and is able 
> write Mathematica procedure. I was obtained that same sequence in 
> completely another problem and connection is very unexpected  I want 
> check that these two are that same. My procedure is on number of factors 
> of  polynomials (x^n+x+1) :
> Table[Length[FactorList[x^n + x + 1]] - 1, {n, 0, 200}]
> {1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 
> 1,
> 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 
> 2, 1,
> 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 
> 1, 2,
> 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 
> 1, 1,
> 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 
> 2, 1,
> 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 
> 1, 2,
> 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 
> 1, 1,
> 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2}
>
> I will be greatfull for procedure or explanation that these two different 
> matters are that same
>
> Best wishes
> Artur
>