Practical number
such that all smaller positive integers can be represented as sums of distinct divisors of
Fibonacci does not formally define practical numbers, but he gives a table of Egyptian fraction expansions for fractions with practical denominators.
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."
His partial classification of these numbers was completed by Stewart (1954) and Sierpiński (1955).
This characterization makes it possible to determine whether a number is practical by examining its prime factorization.
If the ordered set of all divisors of the practical number
This partial characterization was extended and completed by Stewart (1954) and Sierpiński (1955) 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
The condition stated above is necessary and sufficient for a number to be practical.
, 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.
, by the following sequence of steps:[4] Several other notable sets of integers consist only of practical numbers: If
Fibonacci, in his 1202 book Liber Abaci[2] 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.
The proof involves finding a sequence of practical numbers
Expanding both numerators on the right hand side of this formula into sums of divisors of
results in the desired Egyptian fraction representation.
has an Egyptian fraction representation in which the largest denominator is
According to a September 2015 conjecture by Zhi-Wei Sun,[8] every positive rational number has an Egyptian fraction representation in which every denominator is a practical number.
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
[9] Melfi also showed[10] that there are infinitely many practical Fibonacci numbers (sequence A124105 in the OEIS); the analogous question of the existence of infinitely many Fibonacci primes is open.
Hausman & Shapiro (1984) showed that there always exists a practical number in the interval
, a result analogous to Legendre's conjecture for primes.
, a formula which resembles the prime number theorem, strengthening the earlier claim of Erdős & Loxton (1979) that the practical numbers have density zero in the integers.
Improving on an estimate of Tenenbaum (1986), Saias (1997) found that
The Erdős–Kac theorem implies that for a large random integer
(counted with or without multiplicity) follows an approximate normal distribution with mean
, the number of prime factors is approximately normal with mean
