On a question of Douglass and Ono

It is known that the partition function $p(n)$ obeys Benford's law in any integer base $b\ge 2$. A similar result was obtained by Douglass and Ono for the plane partition function $\text{PL}(n)$ in a recent paper. In their paper, Douglass and Ono asked for an explicit version of this result. In particular, given an integer base $b\ge 2$ and string $f$ of digits in base $b$ they asked for an explicit value $N(b,f)$ such that there exists $n\le N(b,f)$ with the property that $\text{PL}(n)$ starts with the string $f$ when written in base $b$. In my talk, I will present an explicit value for $N(b,f)$ both for the partition function $p(n)$ as well as for the plane partition function $\text{PL}(n)$.