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Published online by Cambridge University Press: 24 October 2018
Let 
$k$ be an arbitrary positive integer and let 
$\unicode[STIX]{x1D6FE}(n)$ stand for the product of the distinct prime factors of 
$n$. For each integer 
$n\geqslant 2$, let 
$a_{n}$ and 
$b_{n}$ stand respectively for the maximum and the minimum of the 
$k$ integers 
$\unicode[STIX]{x1D6FE}(n+1),\unicode[STIX]{x1D6FE}(n+2),\ldots ,\unicode[STIX]{x1D6FE}(n+k)$. We show that 
$\liminf _{n\rightarrow \infty }a_{n}/b_{n}=1$. We also prove that the same result holds in the case of the Euler function and the sum of the divisors function, as well as the functions 
$\unicode[STIX]{x1D714}(n)$ and 
$\unicode[STIX]{x1D6FA}(n)$, which stand respectively for the number of distinct prime factors of 
$n$ and the total number of prime factors of 
$n$ counting their multiplicity.
The work of the first author was supported in part by a grant from NSERC of Canada. The work of the second author was supported in part by grant CPRR160325161141 and an A-rated scientist award, both from the NRF of South Africa, and by grant no. 17-02804S of the Czech Granting Agency.
