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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 $\,m\ge 2,$ $\,m=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_\ell^{\alpha_\ell},$ with primes $p_1<p_2<\dots<p_\ell$ and integers $\alpha_i\ge 1,$ is practical if and only if $\,p_1=2\,$ and, for $i=2,3,\dots,\ell,$ $$p_i\le\sigma\!\left(p_1^{\alpha_ 1}p_2^{\alpha_2}\cdots p_{i-1} ^{\alpha_{i-1}}\right)+1,$$ 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], proved a good estimate in terms of a Chebishev-type theorem: for suitable constants $c_1$ and $c_2$ , $$c_1\frac x{\log x}<P(x)<c_2\frac x{\log x}.$$
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 $m-2,m,m+2$ .
- 1
- G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996), 205-210.
- 2
- E. Saias, Entiers à diviseurs denses 1, J. Number Theory 62 (1997), 163-191.
- 3
- G. Tenenbaum, Sur un problème de crible et ses applications, Ann. Sci. Éc. Norm. Sup. (4) 19 (1986), 1-30.
- 4
- G. Tenenbaum, Sur un problème de crible et ses applications, 2. Corrigendum et étude du graphe divisoriel, Ann. Sci. Éc. Norm. Sup. (4) 28 (1995), 115-127.
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