least integer for a related factorial

Question

Let $p$ be a prime, $d$ a divisor of $p-1$, $j$ an integer with $0 \le j<d$. Let $n\in\mathbb N$. Does there exist a formula giving the least integer $m$ such that $$v_p((dm)!)\ge v_p((dn)!)+j+dn?$$ Thanks in advance

Answer 1

Your question is equivalent to $$ dm - s(dm) \geq pdn -(p-1)s(dn)+(p-1)j $$ where $s$ is the sum of digits to base $p$. Since $s$ is rather small, for large $n$ we have that $m$ is pretty close to $pn$. Moreover, $s(dpn)=s(dn)$, thus $m\leq dpn$, unless $(p-2)s(dn)<(p-1)j$. However, whether the inequality holds for $m=pn-1$ depends in a rather complicated way on the digits of $pn$, so there is no simple exact expression.