For integers $n \geq k \geq 1$ let $$H(n,k) := \sum_{1 \leq i_1 < \cdots < i_k \leq n} \frac1{i_1 \cdots i_k}$$ be the $k$-th elementary symmetric function of $\tfrac1{1},\tfrac1{2}, \ldots, \tfrac1{n}$.

Erdős and Niven [1] proved that $H(n, k)$ is an integer only for finitely many $n$ and $k$, and subsequently Chen and Tang [2] showed that $H(1,1)$ and $H(3,2)$ are the only integral values.

**My question is:** *"Is it true that $H(n,k)$ is a $2$-adic integer only for finitely many $n$ and $k$?"*

Note the two extremal cases: $H(n, 1) = 1 + \frac1{2} + \cdots + \frac1{n}$, the $n$-th harmonic number, which is well-known to be a $2$-adic integer only for $n = 1$; and $H(n,n) = 1 / n!$, which obviously is a $2$-adic integer only for $n = 1$. Note also that the $p$-adic valuation of $H(n,k)$ has been studied in [3].

Of course one may ask the more general question: *"Given a prime number $p$, is it true that $H(n,k)$ is a $p$-adic integer only for finitely many $n$ and $k$?"* However, an old and still open conjecture of Eswarathasan and Levine [4] states that for any prime $p$ the harmonic number $H(n,1)$ is a $p$-adic integer only for finitely many positive integer $n$. Hence, this latter question seems to be too difficult for the current methods.

[1] P. Erdős and I. Niven, *Some properties of partial sums of the harmonic series*, Bull. Amer. Math. Soc., **52** (1946), 248–251.

[2] Y.-G. Chen and M. Tang, *On the elementary symmetric functions of $1, 1/2, . . . , 1/n$*, Amer. Math. Monthly, **119** (2012), 862–867.

[3] P. Leonetti and C. Sanna, *On the p-adic valuation of Stirling numbers of the first kind*, Acta Mathematica Hungarica **151** (2017), 217–231.

[4] A. Eswarathasan and E. Levine, *$p$-integral harmonic sums*, Discrete Math., **91** (1991), 249–257.

gpcode:f(n,p)=p=prod(i=1,n,1+x/i); vector(n,j,-valuation(polcoeff(p,j),2)); vector(256,n,f(n))$\endgroup$ – Noam D. Elkies Mar 8 '17 at 2:262more comments