practical number

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

.An integer $m\ge 2,$$m=p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\mathrm{\cdots }{p}_{\mathrm{\ell }}^{{\alpha }_{\mathrm{\ell }}},$with primes and integers ${\alpha }_i\ge 1,$is practical if and only if $p_1=2$ and, for$i=2,3,\mathrm{\dots },\mathrm{\ell },$

$$p_i\le \sigma (p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\mathrm{\cdots }{p}_{i-1}{}^{{\alpha }_{i-1}})+1,$$

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], proveda good estimate in terms ofa Chebishev-type theorem: for suitableconstants $c_1$ and $c_2$,

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