In number theory, a practical number or panarithmic number is a positive integer such that all smaller positive integers can be represented as sums of distinct divisors of . 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 (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.
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." His partial classification of these numbers 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 (including 1 and itself) 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 (with the primes in sorted order ) 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(2) + 1 = 4, 29 ≤ σ(2 × 32) + 1 = 40, and 823 ≤ σ(2 × 32 × 29) + 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 , 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 satisfies the condition above, then any can be represented as a sum of divisors of , by the following sequence of steps: By induction on , it can be shown that . Hence . Since the internals cover for , there are such a and some such that . Since and can be shown by induction to be practical, we can find a representation of q as a sum of divisors of . Since , and since can be shown by induction to be practical, we can find a representation of r as a sum of divisors of . The divisors representing r, together with times each of the divisors representing q, together form a representation of m as a sum of divisors of .