In number theory, a practical number or panarithmic number is a positive integer n {\displaystyle n} such that all smaller positive integers can be represented as sums of distinct divisors of n {\displaystyle 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 (sequence A005153 in the 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. The name "practical number" is due to Srinivasan (1948). 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. 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 Srinivasan (1948) 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 n {\displaystyle n} is d 1 , d 2 , . . . , d j {\displaystyle {d_{1},d_{2},...,d_{j}}} with d 1 = 1 {\displaystyle d_{1}=1} and d j = n {\displaystyle d_{j}=n} , then Srinivasan's statement can be expressed by the inequality
2 n ≤ 1 + ∑ i = 1 j d i . {\displaystyle 2n\leq 1+\sum _{i=1}^{j}d_{i}.}
In other words, the ordered sequence of all divisors d 1 < d 2 < . . . < d j {\displaystyle {d_{1}<d_{2}<...<d_{j}}} of a practical number has to be a complete sub-sequence. 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 n = p 1 α 1 . . . p k α k {\displaystyle n=p_{1}^{\alpha _{1}}...p_{k}^{\alpha _{k}}} (with the primes in sorted order p 1 < p 2 < ⋯ < p k {\displaystyle p_{1}<p_{2}<\dots <p_{k}} ) is practical if and only if each of its prime factors p i {\displaystyle p_{i}} is small enough for p i − 1 {\displaystyle p_{i}-1} to have a representation as a sum of smaller divisors. For this to be true, the first prime p 1 {\displaystyle p_{1}} must equal 2 and, for every i from 2 to k, each successive prime p i {\displaystyle p_{i}} must obey the inequality
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