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practical numbers
A number    is called practical if all the numbers   can be written as the sum of distinct proper divisors of  .

For example, 18 is practical because every smaller number can be written as a sum of its proper divisors, 1, 2, 3, 6, 9, like 13=1+3+9.

Steward and SierpiĆski have characterized completely the set of practical numbers as follows. A number  , whose prime factorization is    is a practical number if and only if it is even (i.e.,  ) and, for every  , it holds

where    denotes the sum of divisors of  .

The first practical numbers are 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, 156, 160 more terms

Below, the spiral pattern of practical numbers up to  . See the page on prime numbers for an explanation and links to similar pictures.

Practical numbers can also be... (you may click on names or numbers and on + to get more values)