Practical Numbers

A practical number n is a positive integer such that all smaller positive integers can be written as sums of distinct divisors of n. This means that, for example, if you are making up a set of weights to use on a balance scale, you need only weights equal to the divisors of the largest weight to balance any weight from 1 to the largest.

For example, 20 is a primitive practical number, since a set of weights composed of 20, 10, 5, 4, 2, and 1 units can express even weights of any value from 1 to 20 using no more than one of each weight in the set. This is said to explain why many traditional units of weight, length, volume, and money were divided into sub-units with numbers like 12 (inches per foot, hours per half-day, pence per British shilling), 16 (ounces per pound), 20 (shillings per British pound), 24 (hours per day), 28 (words per FASTRAND sector), and 5280 (feet per mile). Practical numbers, having many divisors, simplify measurements of subdivisions in mental arithmetic. Interestingly, 10 is not a practical number, so it can be argued that replacement of traditional units with decimal systems is “impractical”.

All powers of two are practical numbers, and the product of two practical numbers is also a practical number and hence the set of practical numbers is infinite. The practical numbers are sequence A005153 in the On-Line Encyclopedia of Integer Sequences and the subset of primitive practical numbers is sequence A267124.