No abstract

No abstract

In 1862, Wolstenholme proved that the numerator of the [Formula: see text]th harmonic number is divisible by [Formula: see text] for any prime [Formula: see text]. A variation of this theorem was shown by Alkan and Leudesdorf. Motivated by these results, we prove a congruence modulo some odd primes for some generalized harmonic type sums.

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We show that there exist infinitely many hyperharmonic integers, and this refutes a conjecture of Mező. In particular, for r = 64•(2 α −1)+32, the hyperharmonic number h (r ) 33 is integer for 153 different values of α (mod 748 440), where the smallest r is equal to 64 • (2 2659 − 1) + 32.