practicalnumber

A practical number is a positive integer n with the property that every integer m between 1 and n can be written as a sum of distinct divisors of n. In other words, the divisors of n are sufficient to build all smaller positive integers by selecting a subset of them.

A useful characterization of practical numbers uses prime factorization. If n = ∏ p_i^{a_i} with primes p_1 < p_2 < ... < p_k, then n is practical if and only if p_1 = 2 and for each i = 1, 2, ..., k−1 we have p_{i+1} ≤ 1 + σ(∏_{j=1}^{i} p_j^{a_j}), where σ denotes the sum of divisors function. This condition gives a constructive procedure for checking practicality based on the ordered prime factors and their powers.

Elementary properties include that every practical number greater than 1 is even, and 1 is usually treated as practical. The property is preserved under doubling: if n is practical, then 2n is practical. The sequence of practical numbers begins 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 60, and so on.

Distribution and research: practical numbers have been studied in analytic and combinatorial number theory, with ongoing questions about their density and distribution. They are cataloged in the OEIS as A005153 and are related to the study of divisor sums and representations by subsets of divisors.

See also: divisor function, sums of divisors, practical numbers in number theory literature. References: OEIS sequence A005153.