Curiosity about a simple identity Consider the well known formula
$$1^3 + 2^3 +\cdots+ n^3 = (1+\cdots+n)^2  , n \in N$$
Now suppose that for all $n \in N$ the identity is true :
$$1^k + 2^k +\cdots+ n^k = (1+\cdots+n)^{k-1}  $$
with $k \in N$ fixed. Which is the possible values of $k$ ?
Someone could point me a reference?
Sorry for my english, it is not good
Thanks in advance

The dominant term of the left hand side is  $$ \frac{1}{k+1} n^{k+1}.  $$
The right hand side is $$  \left( \frac{n^2 + n}{2} \right)^{k-1},  $$ dominant term
$$ \frac{2}{2^k} n^{2k - 2}.  $$
So $$  k+1 = 2k - 2 $$ and
$$  3 = k $$
