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| In [[number theory]], a '''practical number''' or '''panarithmic number'''<ref>{{harvtxt|Margenstern|1991}} cites {{harvtxt|Robinson|1979}} and {{harvtxt|Heyworth|1980}} for the name "panarithmic numbers".</ref> is a positive integer ''n'' such that all smaller positive integers can be represented as sums of distinct [[divisor]]s of ''n''. For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6: as well as these divisors themselves, we have 5 = 3 + 2, 7 = 6 + 1, 8 = 6 + 2, 9 = 6 + 3, 10 = 6 + 3 + 1, and 11 = 6 + 3 + 2.
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| The sequence of practical numbers {{OEIS|A005153}} begins
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| :1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, ....
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| Practical numbers were used by [[Fibonacci]] in his [[Liber Abaci]] (1202) in connection with the problem of representing rational numbers as [[Egyptian fraction]]s. Fibonacci does not formally define practical numbers, but he gives a table of Egyptian fraction expansions for fractions with practical denominators.<ref name="sigler">{{harvtxt|Sigler|2002}}.</ref>
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| The name "practical number" is due to {{harvtxt|Srinivasan|1948}}, who first attempted a classification of these numbers that was completed by {{harvtxt|Stewart|1954}} and {{harvtxt|Sierpiński|1955}}. This characterization makes it possible to determine whether a number is practical by examining its prime factorization. Any even [[perfect number]] and any [[power of two]] is also a practical number.
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| Practical numbers have also been shown to be analogous with [[prime number]]s in many of their properties.<ref>{{harvtxt|Hausman|Shapiro|1984}}; {{harvtxt|Margenstern|1991}}; {{harvtxt|Melfi|1996}}; {{harvtxt|Saias|1997}}.</ref>
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| ==Characterization of practical numbers==
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| As {{harvtxt|Stewart|1954}} and {{harvtxt|Sierpiński|1955}} showed, it is straightforward to determine whether a number is practical from its [[prime factorization]].
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| A positive integer <math>n=p_1^{\alpha_1}...p_k^{\alpha_k}</math> with <math>n>1</math> and primes <math>p_1<p_2<\dots<p_k</math> is practical if and only if <math>p_1=2</math> and, for every ''i'' from 2 to ''k'',
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| :<math>p_i\leq1+\sigma(p_1^{\alpha_1}\dots p_{i-1}^{\alpha_{i-1}})=1+\prod_{j=1}^{i-1}\frac{p_j^{\alpha_j+1}-1}{p_j-1},</math>
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| where <math>\sigma(x)</math> denotes the [[Divisor function|sum of the divisors]] of ''x''. For example, 3 ≤ σ(2)+1 = 4, 29 ≤ σ(2 × 3<sup>2</sup>)+1 = 40, and 823 ≤ σ(2 × 3<sup>2</sup> × 29)+1=1171, so 2 × 3<sup>2</sup> × 29 × 823 = 429606 is practical. This characterization extends a partial classification of the practical numbers given by {{harvtxt|Srinivasan|1948}}.
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| It is not difficult to prove that this condition is necessary and sufficient for a number to be practical. In one direction, this condition is clearly necessary in order to be able to represent <math>p_i-1</math> as a sum of divisors of ''n''. In the other direction, the condition is sufficient, as can be shown by induction. More strongly, one can show that, if the factorization of ''n'' satisfies the condition above, then any <math>m \le \sigma(n)</math> can be represented as a sum of divisors of ''n'', by the following sequence of steps:
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| * Let <math>q = \min\{\lfloor m/p_k^{\alpha_k}\rfloor, \sigma(n/p_k^{\alpha_k})\}</math>, and let <math>r = m - qp_k^{\sigma_k}</math>.
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| * Since <math>q\le\sigma(n/p_k^{\alpha_k})</math> and <math>n/p_k^{\alpha_k}</math> can be shown by induction to be practical, we can find a representation of ''q'' as a sum of divisors of <math>n/p_k^{\alpha_k}</math>.
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| * Since <math>r\le \sigma(n) - p_k^{\alpha_k}\sigma(n/p_k^{\alpha_k}) = \sigma(n/p_k)</math>, and since <math>n/p_k</math> can be shown by induction to be practical, we can find a representation of ''r'' as a sum of divisors of <math>n/p_k</math>.
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| * The divisors representing ''r'', together with <math>p_k^{\alpha_k}</math> times each of the divisors representing ''q'', together form a representation of ''m'' as a sum of divisors of ''n''.
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| ==Relation to other classes of numbers==
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| Any [[power of two]] is a practical number.<ref name="s48"/> Powers of two trivially satisfy the characterization of practical numbers in terms of their prime factorizations: the only prime in their factorizations, ''p''<sub>1</sub>, equals two as required. Any even [[perfect number]] is also a practical number:<ref name="s48"/> due to [[Euler]]'s result that these numbers must have the form 2<sup>''n'' − 1</sup>(2<sup>''n''</sup> − 1), every odd prime factor of an even perfect number must be at most the sum of the divisors of the even part of the number, and therefore the number must satisfy the characterization of practical numbers.
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| Any [[primorial]] is practical.<ref name="s48"/> By [[Bertrand's postulate]], each successive prime in the prime factorization of a primorial must be smaller than the product of the first and last primes in the factorization of the preceding primorial, so primorials necessarily satisfy the characterization of practical numbers. Therefore, also, any number that is the product of nonzero powers of the first ''k'' primes must also be practical; this includes [[Ramanujan]]'s [[highly composite number]]s (numbers with more divisors than any smaller positive integer) as well as the [[factorial]] numbers.<ref name="s48">{{harvtxt|Srinivasan|1948}}.</ref>
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| ==Practical numbers and Egyptian fractions==
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| If ''n'' is practical, then any [[rational number]] of the form ''m''/''n'' may be represented as a sum ∑''d<sub>i</sub>''/''n'' where each ''d<sub>i</sub>'' is a distinct divisor of ''n''. Each term in this sum simplifies to a [[unit fraction]], so such a sum provides a representation of ''m''/''n'' as an [[Egyptian fraction]]. For instance,
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| :<math>\frac{13}{20}=\frac{10}{20}+\frac{2}{20}+\frac{1}{20}=\frac12+\frac1{10}+\frac1{20}.</math>
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| Fibonacci, in his 1202 book ''[[Liber Abaci]]''<ref name="sigler"/> lists several methods for finding Egyptian fraction representations of a rational number. Of these, the first is to test whether the number is itself already a unit fraction, but the second is to search for a representation of the numerator as a sum of divisors of the denominator, as described above; this method is only guaranteed to succeed for denominators that are practical. Fibonacci provides tables of these representations for fractions having as denominators the practical numbers 6, 8, 12, 20, 24, 60, and 100.
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| {{harvtxt|Vose|1985}} showed that every number ''x''/''y'' has an Egyptian fraction representation with <math>\scriptstyle O(\sqrt{\log y})</math> terms. The proof involves finding a sequence of practical numbers ''n''<sub>''i''</sub> with the property that every number less than ''n''<sub>''i''</sub> may be written as a sum of <math>\scriptstyle O(\sqrt{\log n_{i-1}})</math> distinct divisors of ''n''<sub>''i''</sub>. Then, ''i'' is chosen so that ''n''<sub>''i'' − 1</sub> < ''y'' ≤ ''n''<sub>''i''</sub>, and ''xn<sub>i</sub>'' is divided by ''y'' giving quotient ''q'' and remainder ''r''. It follows from these choices that <math>\scriptstyle\frac{x}{y}=\frac{q}{n_i}+\frac{r}{yn_i}</math>. Expanding both numerators on the right hand side of this formula into sums of divisors of ''n''<sub>''i''</sub> results in the desired Egyptian fraction representation. {{harvtxt|Tenenbaum|Yokota|1990}} use a similar technique involving a different sequence of practical numbers to show that every number ''x''/''y'' has an Egyptian fraction representation in which the largest denominator is <math>\scriptstyle O(\frac{y\log^2 y}{\log\log y})</math>.
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| == Analogies with prime numbers ==
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| One reason for interest in practical numbers is that many of their properties are similar to properties of the [[prime numbers]]. For example, if ''p''(''x'') is the enumerating function of practical numbers, i.e., the number of practical numbers not exceeding ''x'', {{harvtxt|Saias|1997}} proved that for suitable constants ''c''<sub>1</sub> and ''c''<sub>2</sub>:
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| :<math>c_1\frac x{\log x}<p(x)<c_2\frac x{\log x},</math>
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| a formula which resembles the [[prime number theorem]]. This result largely resolved a conjecture of {{harvtxt|Margenstern|1991}} that ''p''(''x'') is asymptotic to ''cx''/log ''x'' for some constant ''c'', and it strengthens an earlier claim of {{harvtxt|Erdős|Loxton|1979}} that the practical numbers have density zero in the integers. | |
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| Theorems analogous to [[Goldbach's conjecture]] and the [[twin prime conjecture]] are also known for practical numbers: every positive even integer is the sum of two practical numbers, and there exist infinitely many triples of practical numbers ''x'' − 2, ''x'', ''x'' + 2.<ref>{{harvtxt|Melfi|1996}}.</ref> [[Giuseppe Melfi|Melfi]] also showed that there are infinitely many practical [[Fibonacci number]]s {{OEIS|id=A124105}}; the analogous question of the existence of infinitely many [[Fibonacci prime]]s is open. {{harvtxt|Hausman|Shapiro|1984}} showed that there always exists a practical number in the interval [''x''<sup>2</sup>,(''x'' + 1)<sup>2</sup>] for any positive real ''x'', a result analogous to [[Legendre's conjecture]] for primes.
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| ==Notes==
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| == External links ==
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| *[http://www.dm.unipi.it/gauss-pages/melfi/public_html/pratica.html Tables of practical numbers] compiled by Giuseppe Melfi.
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| *{{PlanetMath |urlname=PracticalNumber |title=Practical Number}}
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| *{{Mathworld |urlname=PracticalNumber |title=Practical Number}}
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| {{Divisor classes}}
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| {{Classes of natural numbers}}
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| [[Category:Integer sequences]]
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| [[Category:Egyptian fractions]]
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