Does there exist an odd prime $p>3$, and integers $m>2$ and $n_k>0$ such that

$$ p^{\sum_{k=1}^m n_k} \equiv 1 \bmod \prod_{k=1}^m (p^{n_k}-1) ? $$

where if some n_i are the same , the same number appears at most r(n,p) times. r(n,p) is the number of irreducible polynomials of degree n over the finite field F_p.