On certain properties of harmonic numbers

On certain properties of harmonic numbers. Let Hn be the n -th harmonic number and let un be its numerator. For any prime p , let Jp be the set of positive integers n with p|un. In 1991, Eswarathasan and Levine conjectured that Jp is finite for any prime p. It is clear that the p -adic valuation of Hn is not less than −⌊logp⁡n⌋. Let Tp be the set of positive integers n such that the p -adic valuation of Hn is equal to −⌊logp⁡n⌋.