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

R J Cano 0xcc00ffffeeee at gmail.com
Fri Mar 16 06:18:05 CET 2018

```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?

At purpose, as an example candidate for such kind of function g(n),
particular case could be n=9, there: n+1=10; A062813(10)=9876543210;
and A023811(10)=123456789; So: g(9)= 1219326311126352700 ( which of
course is 0 mod 100 )

Also as a related issue and a more simple sequence than the one
proposed here initially, we could ask for: "Least integer k such that
k is squarefree and n+k is 0 mod (n+1)^2"; ... well, this other
sequence prepended by an 1 match its first eighteen terms with A002061
(http://oeis.org/A002061) where the 19th term is 343=7^3 whereas it
would be expected 1065; Indeed the first 33 terms are: 1, 3, 7, 13,
21, 31, 43, 57, 73, 91, 111, 133, 157, 183, 211, 241, 273, 307,
[1065], 1181, 1303, 1915, 2094, 2281, 3101, 3355, 3619, 3893, 4177,
4471, 6697, 8161, 9769, ...; And this last one is not already present
in the OEIS  (Note: There 1 was interpreted as the zeroth squarefree
number when sequencing that).