On the number of numerical semigroups of prime power genus

Given $g\ge 1$, the number $n(g)$ of numerical semigroups $S \subset \N$ of genus $|\N \setminus S|$ equal to $g$ is the subject of challenging conjectures of Bras-Amor\'os. In this paper, we focus on the counting function $n(g,2)$ of \textit{two-generator} numerical semigroups of genus $g$, which is known to also count certain special factorizations of $2g$. Further focusing on the case $g=p^k$ for any odd prime $p$ and $k \ge 1$, we show that $n(p^k,2)$ only depends on the class of $p$ modulo a certain explicit modulus $M(k)$. The main ingredient is a reduction of $\gcd(p^\alpha+1, 2p^\beta+1)$ to a simpler form, using the continued fraction of $\alpha/\beta$. We treat the case $k=9$ in detail and show explicitly how $n(p^9,2)$ depends on the class of $p$ mod $M(9)=3 \cdot 5 \cdot 11 \cdot 17 \cdot 43 \cdot 257$