Arranging Consecutive Integers in a Continued Radical to Make a Rational Number

Find the sum of all positive integral $n$ such that there exists a permutation $\sigma$ of the set $\{ 1,\ldots, n\}$ such that

$\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\cdots+\sqrt{\sigma(n)}}}}$

is a rational number. $\sigma(m)$ denotes the $m$ -th term of the permutation.