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.
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