[seqfan] Re: On squarefree numbers sequences obtained by operating with permutations.

[Max Alekseyev]

On Fri, Mar 16, 2018 at 1:18 AM, R J Cano <0xcc00ffffeeee at gmail.com> wrote:

> Dear Seqfan,
>
> I'm interested on figuring out how to prove that: IF it is given a
> mapping between nonnegative integers "g : N U {0} ---> S" ( where S is
> a subset of N U {0} ), THEN the fact of having an explicit
> (polynomial) formula for g(n) such that "n+g(n)= 0 (mod) (n+1)^2" is a
> sufficient condition for ensuring that g(n) is squarefree for n>0;
>
> Could someone give some advise in this matter?
>
>
It's not sufficient.
For example, take g(n) = (n+1)^2 - n = n^2 + n + 1, which by definition
satisfies n+g(n) == 0 (mod (n+1)^2). However, g(18) = 343 = 7^3 is
not squarefree.

Regards,
Max