Carmichael numbers are known to have the property $p-1 | N-1$ for all $p | N$. It is also easy to find numbers $N$ such that $p+1 | N+1$ for all $p | N$ (these numbers lead to strong lucas pseduoprimes). Are there any examples of $p^2+1 | N^2+1$ for all $p | n$? I didn't find any prime $p < 1000$. Even without the prime restriction, I still wasn't able to find any integers with this property. The question is, find a set of integers $>1$ (not necessarily prime) $[a_1, a_2, a_3,...a_n]$ where $N=a_1*a_2*a_3*...*a_n$ and $(a_k^2+1) | (N^2+1)$ for all $k$?
Note: $N ≠ p$ and $N$ cannot be prime.