Journal of Integer Sequences, Vol. 14 (2011), Article 11.7.2

The 4-Nicol Numbers Having Five Different Prime Divisors

Qiao-Xiao Jin and Min Tang
Department of Mathematics
Anhui Normal University
Wuhu 241000
P. R. China


A positive integer $ n$ is called a Nicol number if $ n\mid
\varphi(n)+\sigma(n)$, and a t-Nicol number if $ \varphi(n)+\sigma(n)=tn$. In this paper, we show that if $ n$ is a 4-Nicol number that has five different prime divisors, then $ n=2^{\alpha_{1}}\cdot 3\cdot 5^{\alpha_{3}}\cdot p^{\alpha_{4}}\cdot q^{\alpha_{5}}$, or $ n=2^{\alpha_{1}}\cdot 3\cdot 7^{\alpha_{3}}\cdot p^{\alpha_{4}}\cdot q^{\alpha_{5}}$ with $ p\leq 29$.

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Received April 12 2011; revised version received July 7 2011. Published in Journal of Integer Sequences, September 4 2011. Revised, April 11 2012.

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