$1^{k}+2^{k}+\cdots+n^{k}$ is not divisible by $n+2$ $k$ is odd number.
show that for arbitrary $n\in N$ , $1^{k}+2^{k}+\cdots+n^{k}$ is not divisible by $n+2$    

As $k$ is odd, $r^k+(n+2-r)^k$ is divisible by $r+n+2-r=n+2$
Putting $r=1,2,\cdots,n,n+1$ and adding we get,
$2\{1^k+2^k+\cdots +n^k+(n+1)^k\}$ is divisible by $n+2$
If $(n+2)$ divides $(1^k+2^k+\cdots +n^k),$ 
it will divide $2\{1^k+2^k+\cdots +n^k+(n+1)^k\}-2(1^k+2^k+\cdots +n^k)=2(n+1)^k$
