Practical number


In number theory, 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.
The sequence of practical numbers begins
Practical numbers were used by Fibonacci in his Liber Abaci 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.
The name "practical number" is due to . He noted that "the subdivisions of money, weights, and measures involve numbers like 4, 12, 16, 20 and 28 which are usually supposed to be so inconvenient as to deserve replacement by powers of 10." He rediscovered the number theoretical property of such numbers and was the first to attempt a classification of these numbers that was completed by and. This characterization makes it possible to determine whether a number is practical by examining its prime factorization. Every even perfect number and every power of two is also a practical number.
Practical numbers have also been shown to be analogous with prime numbers in many of their properties.

Characterization of practical numbers

The original characterisation by stated that a practical number cannot be a deficient number, that is one of which the sum of all divisors is less than twice the number unless the deficiency is one. If the ordered set of all divisors of the practical number is with and, then Srinivasan's statement can be expressed by the inequality
In other words, the ordered sequence of all divisors of a practical number has to be a complete sub-sequence.
This partial characterization was extended and completed by and who showed that it is straightforward to determine whether a number is practical from its prime factorization.
A positive integer greater than one with prime factorization is practical if and only if each of its prime factors is small enough for to have a representation as a sum of smaller divisors. For this to be true, the first prime must equal 2 and, for every from 2 to , each successive prime must obey the inequality
where denotes the sum of the divisors of x. For example, 2 × 32 × 29 × 823 = 429606 is practical, because the inequality above holds for each of its prime factors: 3 ≤ σ + 1 = 4, 29 ≤ σ + 1 = 40, and 823 ≤ σ + 1 = 1171.
The condition stated above is necessary and sufficient for a number to be practical. In one direction, this condition is necessary in order to be able to represent as a sum of divisors of n, because if the inequality failed to be true then even adding together all the smaller divisors would give a sum too small to reach. In the other direction, the condition is sufficient, as can be shown by induction.
More strongly, if the factorization of n satisfies the condition above, then any can be represented as a sum of divisors of n, by the following sequence of steps:
Several other notable sets of integers consist only of practical numbers:
If n is practical, then any rational number of the form m/n with 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,
Fibonacci, in his 1202 book Liber Abaci 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.
showed that every number x/y has an Egyptian fraction representation with terms. The proof involves finding a sequence of practical numbers ni with the property that every number less than ni may be written as a sum of distinct divisors of ni. Then, i is chosen so that ni − 1 < yni, and xni is divided by y giving quotient q and remainder r. It follows from these choices that. Expanding both numerators on the right hand side of this formula into sums of divisors of ni results in the desired Egyptian fraction representation. 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.
According to a September 2015 conjecture by Zhi-Wei Sun, every positive rational number has an Egyptian fraction representation in which every denominator is a practical number. There is a proof for the conjecture on David Eppstein's blog.

Analogies with prime numbers

One reason for interest in practical numbers is that many of their properties are similar to properties of the prime numbers.
Indeed, theorems analogous to Goldbach's conjecture and the twin prime conjecture are 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. Melfi also showed that there are infinitely many practical Fibonacci numbers ; the analogous question of the existence of infinitely many Fibonacci primes is open. showed that there always exists a practical number in the interval for any positive real x, a result analogous to Legendre's conjecture for primes. This result on practical numbers on short intervals have subsequently improved by Melfi who proved that if is the sequence of practical numbers, then for sufficiently large n and for a suitable A,
Let p count how many practical numbers are at most x.
conjectured that p is asymptotic to cx/log x for some constant c, a formula which resembles the prime number theorem, strengthening the earlier claim of that the practical numbers have density zero in the integers.
Improving on an estimate of, found that p has order of magnitude x/log x.
proved Margenstern's conjecture by showing that
where Thus the practical numbers are about 33.6% more numerous than the prime numbers. The exact value of the constant factor is given by
where is the Euler–Mascheroni constant and runs over primes.