Practical number
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A practical number or panarithmic number is a positive integer n such that all smaller positive integers can be represented as sums of distinct divisors 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. Any even perfect number and any power of two is also a practical number. The sequence of practical numbers (sequence A005153 in OEIS) begins Practical numbers were used by Fibonacci in his Liber Abaci (1202) in connection with the problem of representing rational numbers as Egyptian fractions. Fibonacci does not formally define practical numbers, but he gives a table of Egyptian fraction expansions for fractions with practical denominators (Sigler 2002). The first occurrence of practical numbers in the modern mathematical literature appears to be by Srinivasan (1948).
Characterization of practical numbersAs Stewart (1954) showed, it is straightforward to determine whether a number is practical from its prime factorization. A positive integer Failed to parse (Missing texvc executable; please see math/README to configure.): n=p_1^{\alpha_1}...p_k^{\alpha_k} with Failed to parse (Missing texvc executable; please see math/README to configure.): n>1 and Failed to parse (Missing texvc executable; please see math/README to configure.): p_1<p_2<\dots<p_k primes is practical if and only if Failed to parse (Missing texvc executable; please see math/README to configure.): p_1=2 and for Failed to parse (Missing texvc executable; please see math/README to configure.): i=2,\dots,k
where Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma(x) denotes the sum of the divisors of x. In one direction, this condition is clearly necessary in order to be able to represent Failed to parse (Missing texvc executable; please see math/README to configure.): p_i-1 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 Failed to parse (Missing texvc executable; please see math/README to configure.): m \le \sigma(n) can be represented as a sum of divisors of n, by the following sequence of steps:
, and let Failed to parse (Missing texvc executable; please see math/README to configure.): r = m - qp_k^{\sigma_k} .
and Failed to parse (Missing texvc executable; please see math/README to configure.): n/p_k^{\alpha_k}
can be shown by induction to be practical, we can find a representation of q as a sum of divisors of Failed to parse (Missing texvc executable; please see math/README to configure.): n/p_k^{\alpha_k}
.
, and since Failed to parse (Missing texvc executable; please see math/README to configure.): n/p_k can be shown by induction to be practical, we can find a representation of r as a sum of divisors of Failed to parse (Missing texvc executable; please see math/README to configure.): n/p_k .
times each of the divisors representing q, together form a representation of m as a sum of divisors of n. For example, 3 ≤ σ(2)+1 = 4, 29 ≤ σ(2 × 32)+1 = 40, and 823 ≤ σ(2 × 32 × 29)+1=1171, so 2 × 32 × 29 × 823 = 429606 is practical. Analogies with prime numbersOne reason for interest in practical numbers is that many of their properties are similar to properties of the prime numbers. For example, if Failed to parse (Missing texvc executable; please see math/README to configure.): p(x) is the enumerating function of practical numbers, i.e., the number of practical numbers not exceeding Failed to parse (Missing texvc executable; please see math/README to configure.): x , Saias (1997) proved that for suitable constants Failed to parse (Missing texvc executable; please see math/README to configure.): c_1 and Failed to parse (Missing texvc executable; please see math/README to configure.): c_2 Failed to parse (Missing texvc executable; please see math/README to configure.): c_1\frac x{\log x}<p(x)<c_2\frac x{\log x},
for some constant c. 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 Failed to parse (Missing texvc executable; please see math/README to configure.): x-2,x,x+2 (Melfi 1996). Melfi also showed that there are infinitely many practical Fibonacci numbers (sequence A124105 in OEIS); the analogous question of the existence of infinitely many Fibonacci primes is open. Hausman and Shapiro (1984) showed that there always exists a practical number in the interval Failed to parse (Missing texvc executable; please see math/README to configure.): [x^2,(x+1)^2] for any positive real x, a result analogous to Legendre's conjecture for primes. Practical numbers and Egyptian fractionsIf n is practical, then any rational number of the form m/n may be represented as a sum ∑di/n where each di 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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