A positive integer $m$ is called a practical number if every positive integer $n<m$ is a sum of distinct positive divisors of $m$.

Lemma. An integer $\,m\geq 2,$ $\,m=p_{1}^{{\alpha_{1}}}p_{2}^{{\alpha_{2}}}\cdots p_{\ell}^{{\alpha_{\ell}}},$ with primes $p_{1}<p_{2}<\dots<p_{\ell}$ and integers $\alpha_{i}\geq 1,$ is practical if and only if $\,p_{1}=2\,$ and, for $i=2,3,\dots,\ell,$

where $\sigma(n)$ denotes the sum of the positive divisors of $n.$

Let $P(x)$ be the counting function of practical numbers. Saias [2], using suitable sieve methods introduced by Tenenbaum [3, 4], proved a good estimate in terms of a Chebishev-type theorem: for suitable constants $c_{1}$ and $c_{2}$,

In [1] Melfi proved a Goldbach-type result showing that every even positive integer is a sum of two practical numbers, and that there exist infinitely many triplets of practical numbers of the form $m-2,m,m+2$.