p-adic valuation of harmonic sums and their connections with Wolstenholme primes

We explore a conjecture posed by Eswarathasan and Levine on the distribution of p-adic valuations of harmonic numbers H(n)=1+1/2+⋯+1/n that states that the set Jp of the positive integers n such that p divides the numerator of H(n) is finite. We proved two results, using a modular-arithmetic approach, one for non-Wolstenholme primes and the other for Wolstenholme primes, on an anomalous asymptotic behaviour of the p-adic valuation of H(pmn) when the p-adic valuation of H(n) equals exactly 3.