On the special harmonic numbers $H_{\lfloor p\mathrm{/}9\rfloor }$ and $H_{\lfloor p\mathrm{/}18\rfloor }$ modulo $p$

Building on work of Zhi-Hong Sun, we establish congruences for the special harmonic numbers $H_{\lfloor }p\mathrm{/}9\rfloor$ and $H_{\lfloor p\mathrm{/}18\rfloor }$ modulo $p$, which contain respectively three and four distinct arithmetic components. We also obtain a complete determination modulo $p$ of the corresponding families of sums of reciprocals of the type studied by Dilcher and Skula. Applications to the first case of Fermat’s Last Theorem are considered.