Odd harmonic numbers exceed 10<sup>24</sup>

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Journal Article
Mathematics of Computation, 2010, 79 (272), pp. 2451 - 2460
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A number n > 1 is harmonic if σ(n) σ nτ(n), where τ(n) and σ(n) are the number of positive divisors of n and their sum, respectively. It is known that there are no odd harmonic numbers up to 1015. We show here that, for any odd number n > 106, τ(n) < n1\3. It follows readily that if n is odd and harmonic, then n > p3a/2 for any prime power divisor pa of n, and we have used this in showing that n > 1018. We subsequently showed that for any odd number n > 9· 1017, τ(n) < n1/4, from which it follows that if n is odd and harmonic, then n > p8a/5 with pa as before, and we use this improved result in showing that n > 1024. © 2010 American Mathematical Society.
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