# Permutations Coprime &...
Edwin Clark
eclark at math.usf.edu
Mon Mar 17 03:15:06 CET 2003
On Sun, 16 Mar 2003, Leroy Quet wrote:
> I have discussed a less-limiting variation of this problem before
> (sans the a(k-1)+a(k+1) coprime to a(k) condition).
>
> For every positive integer n, what are the number of permutations,
> a(1),a(2),a(3),...,a(n),
> of {1,2,3,...,n}
>
> where GCD(a(k),a(k+1)) = 1 for all k, 1<=k<=n-1;
>
> AND where, as well,
> GCD(a(k),a(k-1)+a(k+1)) = 1 for all k, 2<=k<=n-1?
>
> I think the sequence begins:
> 1, 2, 2, 2,...
>
Was this a trick question? :-)
The sequence is
1, 2, 2, 2, 4, 0, 0, 0, 0, . .
with all zeros from n = 6 onward.
Note that you cannot have two consecutive even numbers
and you cannot have "odd, even, odd" in such a permutation.
--Edwin Clark
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