Editing Practical number (section)

{{short description|Number whose sums of distinct divisors represent all smaller numbers}}
[[File:Practical number Cuisenaire rods 12.png|thumb|Demonstration of the practicality of the number 12]]

In [[number theory]], a '''practical number''' or '''panarithmic number'''<ref>{{harvtxt|Margenstern|1991}} cites {{harvtxt|Robinson|1979}} and {{harvtxt|Heyworth|1980}} for the name "panarithmic numbers".</ref> is a positive integer <math>n</math> such that all smaller positive integers can be represented as sums of distinct [[divisor]]s of <math>n</math>. 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 {{OEIS|A005153}} begins
{{bi|left=1.6|1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150....}}

Practical numbers were used by [[Fibonacci]] in his ''[[Liber Abaci]]'' (1202) in connection with the problem of representing rational numbers as [[Egyptian fraction]]s. Fibonacci does not formally define practical numbers, but he gives a table of Egyptian fraction expansions for fractions with practical denominators.<ref name="sigler">{{harvtxt|Sigler|2002}}.</ref>

The name "practical number" is due to  {{harvtxt|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 {{harvtxt|Stewart|1954}} and {{harvtxt|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 number]]s in many of their properties.<ref>{{harvtxt|Hausman|Shapiro|1984}}; {{harvtxt|Margenstern|1991}}; {{harvtxt|Melfi|1996}}; {{harvtxt|Saias|1997}}.</ref>