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arXiv · 1607.02863

The denominators of harmonic numbers (Revised)

Peter Shiu
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Abstract

The denominators $d_n$ of the harmonic number $1+\frac12+\frac13+\cdots+\frac1n$ do not increase monotonically with~$n$. It is conjectured that $d_n=D_n={\rm LCM}(1,2,\ldots,n)$ infinitely often. For an odd prime $p$, the set $\{n:pd_n|D_n\}$ has a harmonic density. Moreover, for $2<p_1<p_2<\cdots<p_k$, with $\log p_1/\log p_i$ ($1\le i\le k$) being linearly independent, there exists $n$ such that $p_1p_2\cdots p_kd_n|D_n$.

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Peter Shiu. 2016-07-11. The denominators of harmonic numbers (Revised). https://arxiv.org/abs/1607.02863

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