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practical number∗
mathcam†
2013-03-21 17:07:21
A positive integer m is called a practical number if every positive integer
n < m is a sum of distinct positive divisors of m.
α`
1 α2
Lemma. An integer m ≥ 2, m = pα
1 p2 · · · p` , with primes p1 < p2 <
· · · < p` and integers αi ≥ 1, is practical if and only if p1 = 2 and, for i =
2, 3, . . . , `,
αi−1 1 α2
pi ≤ σ pα
+ 1,
1 p2 · · · pi−1
where σ(n) denotes the sum of the positive divisors of n.
Let P (x) be the counting function of practical numbers. Saias [?], using
suitable sieve methods introduced by Tenenbaum [?, ?], proved a good estimate
in terms of a Chebishev-type theorem: for suitable constants c1 and c2 ,
c1
x
x
< P (x) < c2
.
log x
log x
In [?] 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.
References
[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.
∗ hPracticalNumberi
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setting: h1i hDefinitioni h11A25i
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