The answer is all natural numbers $n$. For the construction, divide into two cases: $n$ is even: Put pair of numbers that add up to $n+1$ next to each other. To find circular arc whose sum of numbers in that arc is $a(n+1)+b, 0\le b \le n$, start off the arc from $b$, then move to the direction opposite of $b$'s pair, stop after passing exactly $a$ pairs (If $b=0$ simply find the arc covering exactly $a$ pairs). $n$ is odd: Put pair of numbers that add up to $n$ next to each other, and put $n$ by himself and not splitting any other pair. The rest is similar with above, but we do mod $n$ this time instead of $n+1$.