Can you rearrange natural numbers so sum of first $n$ is divisible by $n$ for every $n$? 
Can you rearrange natural numbers so sum of first $n$ is divisible by $n$ for every $n$?

I want to prove this by induction, if we rearrange first $n$ numbers we can always choose an integer so sum is divisible by $n+1$. But how can I prove every number will be used?
Example: $1, 3, 2, 6, 8$ and so on...
Thanks in advance

The greedy algorithm (select the smallest distinct number that works) does work, and you will get OEIS A019444. One of the references there – Venkatachala (2009) – proves that it is an involution (stronger than permutation) of the natural numbers.
