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For any positive integer n, the n-th harmonic number hn is defined as the sum of reciprocals of the first n positive integers. These numbers share several arithmetic properties. For instance, it was shown by Theisinger in 1918 that hn is not an integer, when n>1. Later on, a generalization of harmonic numbers was introduced by Conway and Guy. They defined the n-th hyperharmonic number of order r as hn(r) and it is equal to the sum of the hyperharmonic numbers hk(r-1) where k runs over 1 to n (here, n,r are positive integers, r>1 and hn(1) = hn). Inspired by Theisinger's result, Mező conjectured that there are no hyperharmonic integers except 1. In this talk, we first deal with the bounds on the number of lattice points in the first quadrant that give hyperharmonic integers. As a result, we will deduce that the probability of choosing an (n,r) tuple in [1,x] x [1,x] satisfying the property that hn(r) is an integer, tends to 0, as x tends to infinity. On the contrary, we will show that there are infinitely many different hyperharmonic integers. In particular, with the aid of a computer algebra toolbox, we will obtain a special set of (n,r) tuples which refute Mező's conjecture.