**Characterization of Practical Numbers**

As Stewart (1954) and Sierpiński (1955) showed, it is straightforward to determine whether a number is practical from its prime factorization. A positive integer with and primes is practical if and only if and, for every *i* from 2 to *k*,

where denotes the sum of the divisors of *x*. For example, 3 ≤ σ(2)+1 = 4, 29 ≤ σ(2 × 32)+1 = 40, and 823 ≤ σ(2 × 32 × 29)+1=1171, so 2 × 32 × 29 × 823 = 429606 is practical. This characterization extends a partial classification of the practical numbers given by Srinivasan (1948).

It is not difficult to prove that this condition is necessary and sufficient for a number to be practical. In one direction, this condition is clearly necessary in order to be able to represent as a sum of divisors of *n*. In the other direction, the condition is sufficient, as can be shown by induction. More strongly, one can show that, if the factorization of *n* satisfies the condition above, then any can be represented as a sum of divisors of *n*, by the following sequence of steps:

- Let, and let .
- Since and can be shown by induction to be practical, we can find a representation of
*q*as a sum of divisors of . - Since, and since can be shown by induction to be practical, we can find a representation of
*r*as a sum of divisors of . - The divisors representing
*r*, together with times each of the divisors representing*q*, together form a representation of*m*as a sum of divisors of*n*.

Read more about this topic: Practical Number

### Famous quotes containing the words numbers and/or practical:

“I had a feeling that out there, there were very poor people who didn’t have enough to eat. But they wore wonderfully colored rags and did musical *numbers* up and down the streets together.”

—Jill Robinson (b. 1936)

“Whatever *practical* people may say, this world is, after all, absolutely governed by ideas, and very often by the wildest and most hypothetical ideas. It is a matter of the very greatest importance that our theories of things that seem a long way apart from our daily lives, should be as far as possible true, and as far as possible removed from error.”

—Thomas Henry Huxley (1825–95)