A practical number is one whose divisors are flexible enough that every positive integer below it can be written as a sum of distinct divisors. With 12 (divisors 1, 2, 3, 4, 6, 12) you can compose 5 = 2+3, 7 = 3+4, 11 = 1+4+6 — and every other value up to 12.
The name reflects a practical use: a currency or weight system denominated in the divisors of a practical number can make exact change for any amount. Fibonacci used such reasoning in the Liber Abaci when working with Egyptian fractions.
Stewart's criterion (1954) characterizes practical numbers from their factorization alone: writing the primes in increasing order, each new prime must not exceed one plus the sum of divisors of the product so far — which makes membership fast to test. Like the primes, practical numbers have density \(\sim C n/\log n\), and analogues of Goldbach's conjecture and the twin-prime conjecture are theorems for practical numbers.