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: $$J_p(y)=\{n:\sum _{j=1}^n\frac{y^j}{j}\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}p)\}\mathrm{.}$$ Suppose that $p$ is one of the first 100 odd primes and $y\in \{1,2,\mathrm{.}\mathrm{.}\mathrm{.},p-1\}$, then our calculations prove that $\mathrm{\mid }{J}_p(y)\mathrm{\mid }<\mathrm{\infty }$ in 24240 out of 24578 possible cases. Among other results we show that $\mathrm{\mid }{J}_{13}(9)\mathrm{\mid }=18763$. The paper concludes by discussing some possible applications of our method to sums involving Fibonacci numbers.