AIMS Mathematics, 2017, 2(4): 682-691. doi: 10.3934/Math.2017.4.682.

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On the integrality of the first and second elementary symmetricfunctions of $1, 1/2^{s_2}, ...,1/n^{s_n}$

1 Mathematical College, Sichuan University, Chengdu 610064, P.R. China
2 School of Mathematics and Statistics, Southwest University,Chongqing 400715, P.R. China

It is well known that the harmonic sum $H_{n}(1)=\sum_{1\leq k\leq n}\frac{1}{k}$is never an integer for $n>1$. Erd\"{o}s and Niven proved in 1946 thatthe multiple harmonic sum$H_{n}(\{1\}^r)=\sum_{1\leq k_{1}<\cdots< k_{r}\leq n}\frac{1}{k_{1}\cdots k_{r}}$can take integer values for at most finite many integers $n$. In 2012, Chenand Tang refined this result by showing that $H_{n}(\{1\}^r)$ is an integeronly for $(n,r)=(1,1)$ and $(n,r)=(3,2)$. In this paper, we consider theintegrality problem for the first and second elementary symmetric functionof $1, 1/2^{s_2}, ...,$ $1/n^{s_n}$, we show that none of themis an integer with some natural exceptions.
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Keywords elementary symmetric function; integrality; Bertrand’s postulate; p-adic valuation

Citation: Wanxi Yang, Mao Li, Yulu Feng, Xiao Jiang. On the integrality of the first and second elementary symmetricfunctions of $1, 1/2^{s_2}, ...,1/n^{s_n}$. AIMS Mathematics, 2017, 2(4): 682-691. doi: 10.3934/Math.2017.4.682


  • 1. Y.G. Chen and M. Tang, On the elementary symmetric functions of 1, 1/2,..., 1/n, Am. Math. Mon., 119 (2012), 862-867.
  • 2. P. Erdös and I. Niven, Some properties of partial sums of the harmonic series, B. Am. Math. Soc., 52 (1946), 248-251.    
  • 3. S.F. Hong and C.L. Wang, The elementary symmetric functions of reciprocals of the elements of arithmetic progressions, Acta Math. Hung., 144 (2014), 196-211.    
  • 4. N. Koblitz, p-Adic numbers, p-adic analysis and zeta-functions, GTM 58, Springer-Verlag, New York, 1984.
  • 5. Y.Y. Luo, S.F. Hong, G.Y. Qian and C.L.Wang, The elementary symmetric functions of a reciprocal polynomial sequence, C.R. Math., 352 (2014), 269-272.    
  • 6. M.B. Nathanson, Elementary methods in number theroy, GTM 195, Springer-Verlag, New York, 2000.
  • 7. C.L. Wang and S.F. Hong, On the integrality of the elementary symmetric functions of 1, 1/3,..., 1=(2n - 1), Math. Slovaca, 65 (2015), 957-962.


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