On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients

It is well known that the Bernoulli polynomials $\mathbf{B}_n(x)$ have nonintegral coefficients for $n \geq 1$. However, ten cases are known so far in which the derivative $\mathbf{B}'_n(x)$ has only integral coefficients. One may assume that the number of those derivatives is finite. We can link this conjecture to a recent conjecture about the properties of a product of primes satisfying certain $p$-adic conditions. Using a related result of Bordell\`es, Luca, Moree, and Shparlinski, we then show that the number of those derivatives is indeed finite. Furthermore, we derive other characterizations of the primary conjecture. Subsequently, we extend the results to higher derivatives of the Bernoulli polynomials. This provides a product formula for these denominators, and we show similar finiteness results.