Abstract
For any prime number p, let Jp be the set of positive integers n such that p divides the numerator of the n-th harmonic number Hn. An old conjecture of Eswarathasan and Levine states that Jp is finite. We prove that for x≥1 the number of integers in Jp∩[1,x] is less than 129p2/3x0.765. In particular, Jp has asymptotic density zero. Furthermore, we show that there exists a subset Sp of the positive integers, with logarithmic density greater than 0.273, and such that for any n∈Sp the p-adic valuation of Hn is equal to −⌊logpn⌋.
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