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Let n and k be positive integers with 
$n\ge k+1$ and let 
$\{a_i\}_{i=1}^n$ be a strictly increasing sequence of positive integers. Let 
$S_{n, k}:=\sum _{i=1}^{n-k} {1}/{\mathrm {lcm}(a_{i},a_{i+k})}$. In 1978, Borwein [‘A sum of reciprocals of least common multiples’, Canad. Math. Bull. 20 (1978), 117–118] confirmed a conjecture of Erdős by showing that 
$S_{n,1}\le 1-{1}/{2^{n-1}}$. Hong [‘A sharp upper bound for the sum of reciprocals of least common multiples’, Acta Math. Hungar. 160 (2020), 360–375] improved Borwein’s upper bound to 
$S_{n,1}\le {a_{1}}^{-1}(1-{1}/{2^{n-1}})$ and derived optimal upper bounds for 
$S_{n,2}$ and 
$S_{n,3}$. In this paper, we present a sharp upper bound for 
$S_{n,4}$ and characterise the sequences 
$\{a_i\}_{i=1}^n$ for which the upper bound is attained.
$\boldsymbol {2^n}$
We prove a new irreducibility result for polynomials over 
${\mathbb Q}$ and we use it to construct new infinite families of reciprocal monogenic quintinomials in 
${\mathbb Z}[x]$ of degree 
$2^n$.
