UPPER BOUNDS FOR INFINITARY HARMONIC NUMBERS AND INFINITARY HARMONIOUS TUPLES

Abstract A positive integer N is said to be an infinitary harmonic number (IHN) if the harmonic mean of its infinitary divisors is integral. It is still an open problem whether or not there exist infinitely many IHNs. Cohen and Hagis [‘Infinitary harmonic numbers’, Bull. Aust. Math. Soc. 41 (1989), 151–158] showed that there exist at most finitely many IHNs with a fixed number of I -components. We give an explicit Nielsen-type upper bound for the number of IHNs. We also consider a class of general infinitary amicable numbers and obtain the Borho-type upper bound for them.