Bounds and Computer Searches
W. H. Mills (unpublished; see Muskat) showed that any odd harmonic divisor number above 1 must have a prime power factor greater than 107, and Cohen showed that any such number must have at least three different prime factors. Cohen and Sorli (2010) showed that there are no odd harmonic divisor numbers smaller than 1024.
Cohen, Goto, and others starting with Ore himself have performed computer searches listing all small harmonic divisor numbers. From these results, lists are known of all harmonic divisor numbers up to 2×109, and all harmonic divisor numbers for which the harmonic mean of the divisors is at most 300.
Read more about this topic: Harmonic Divisor Number
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