The Hurwitz equation

The Hurwitz equation is the following diophantine equation.
For fixed $n,k>0$: $$x_1^2+x_2^2+\cdots +x_n^2=k{x}_1{x}_2\dots x_n\text{ for }k,n\in N^{\ast }$$
It can be proved that when this equation has solutions, then $k\le n$.

Can you find all the k such as this equation has solution for $n=61$ ?

Answer format: comma delimited list of $k_s$.

You are given: 1,4,5,13,23 for n = 23
[My timing: 90 sec]

PS: There are much faster solutions than mine ...