Hello,

The question below might be well known, and using different words (I made these up, I'm not a number theorist or specialist in combinatorics)

For all integers $n\geq 2$ denote by $\mathcal{P}(n)$ the set of partitions of $n$, see here for a definition

We say that a postive integer $k$ is ** $\mathbf{n}$-squarable** if there exist $(p_1,\ldots,p_t) \in \mathcal{P}(n)$ such that $k=\sum_{i=1}^t{p_i}^2$.

Let $\alpha(n)=\lfloor (n^2-3n)/4\rfloor$.

I would like to show that the integers $n, n+2, n+4,\ldots ,n+2\alpha(n)$ are $n$-squarable.

If we need to reduce a bit the size of $\alpha(n)$ to make it easier to prove this, I'd happily do that.

Also, given $n$ and $k$ as above, could we find a "constructive" algorithm that would find a partition of $n$ verifying $k=\sum_{i=1}^t{p_i}^2$ (without calculating all elements of $\mathcal{P}(n)$ and then doing a search, obviously).

Thank you.