**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:

“All experience teaches that, whenever there is a great national establishment, employing large *numbers* of officials, the public must be reconciled to support many incompetent men; for such is the favoritism and nepotism always prevailing in the purlieus of these establishments, that some incompetent persons are always admitted, to the exclusion of many of the worthy.”

—Herman Melville (1819–1891)

“Missionaries, whether of philosophy or religion, rarely make rapid way, unless their preachings fall in with the prepossessions of the multitude of shallow thinkers, or can be made to serve as a stalking-horse for the promotion of the *practical* aims of the still larger multitude, who do not profess to think much, but are quite certain they want a great deal.”

—Thomas Henry Huxley (1825–95)