An extension of Boyd's $p$-adic algorithm for the harmonic series

In this paper we will extend a $p$-adic algorithm of Boyd in order to study the size of the set: $$J_p(y)=\left\{n :\sum_{j=1}^{n}\frac{y^j}{j}\equiv 0(\mod p)\right\}.$$ Suppose that $p$ is one of the first 100 odd primes and $y\in\{1,2,...,p-1\}$, then our calculations prove that $|J_p(y)|<\infty$ in 24240 out of 24578 possible cases. Among other results we show that $|J_{13}(9)|=18763$. The paper concludes by discussing some possible applications of our method to sums involving Fibonacci numbers.