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A positive integer  is called a practical number if every positive integer  is a sum of distinct positive divisors of  .
Lemma. An integer 
 with primes
 and integers
 is practical if and only if  and, for
where  denotes the sum of the positive divisors of 
Let  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  and  ,
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  .
- 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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