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practical number (Definition)

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,$

$\displaystyle p_i\le\sigma\!\left(p_1^{\alpha_ 1}p_2^{\alpha_2}\cdots p_{i-1} ^{\alpha_{i-1}}\right)+1,$
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$,

$\displaystyle c_1\frac x{\log x}<P(x)<c_2\frac x{\log x}.$

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$.

Bibliography

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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method for representing rational numbers as sums of unit fractions using practical numbers (Algorithm) by CompositeFan
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Cross-references: triplets, even, terms, estimate, function, primes, divisors, sum, integer, positive
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This is version 3 of practical number, born on 2004-02-27, modified 2004-04-30.
Object id is 5637, canonical name is PracticalNumber.
Accessed 1770 times total.

Classification:
AMS MSC11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)

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