Editing Practical number (section)

==Characterization of practical numbers==
The original characterisation by {{harvtxt|Srinivasan|1948}} stated that a practical number cannot be a [[deficient number]], that is one of which the sum of all divisors (including 1 and itself) is less than twice the number unless the deficiency is one. If the ordered set of all divisors of the practical number <math>n</math> is <math>{d_1, d_2,..., d_j}</math> with <math>d_1=1</math> and <math>d_j=n</math>, then Srinivasan's statement can be expressed by the inequality
<math display=block>2n\leq1+\sum_{i=1}^j d_i.</math>
In other words, the ordered sequence of all divisors <math>{d_1<d_2<...<d_j}</math> of a practical number has to be a [[Complete sequence|complete sub-sequence]].

This partial characterization was extended and completed by {{harvtxt|Stewart|1954}} and {{harvtxt|Sierpiński|1955}} who showed that it is straightforward to determine whether a number is practical from its [[prime factorization]].
A positive integer greater than one with prime factorization <math>n=p_1^{\alpha_1}...p_k^{\alpha_k}</math> (with the primes in sorted order <math>p_1<p_2<\dots<p_k</math>) is practical if and only if each of its prime factors <math>p_i</math> is small enough for <math>p_i-1</math> to have a representation as a sum of smaller divisors. For this to be true, the first prime <math>p_1</math> must equal 2 and, for every {{mvar|i}} from 2 to&nbsp;{{mvar|k}}, each successive prime <math>p_i</math> must obey the inequality
:<math>p_i\leq1+\sigma(p_1^{\alpha_1}p_2^{\alpha_2}\dots p_{i-1}^{\alpha_{i-1}})=1+\sigma(p_1^{\alpha_1})\sigma(p_2^{\alpha_2})\dots \sigma(p_{i-1}^{\alpha_{i-1}})=1+\prod_{j=1}^{i-1}\frac{p_j^{\alpha_j+1}-1}{p_j-1},</math>
where <math>\sigma(x)</math> denotes the [[Divisor function|sum of the divisors]] of ''x''. For example, 2 × 3<sup>2</sup> × 29 × 823 = 429606 is practical, because the inequality above holds for each of its prime factors: 3 ≤ σ(2) + 1 = 4, 29 ≤ σ(2 × 3<sup>2</sup>) + 1 = 40, and 823 ≤ σ(2 × 3<sup>2</sup> × 29) + 1 = 1171.

The condition stated above is necessary and sufficient for a number to be practical. In one direction, this condition is necessary in order to be able to represent <math>p_i-1</math> as a sum of divisors of <math>n</math>, because if the inequality failed to be true then even adding together all the smaller divisors would give a sum too small to reach <math>p_i-1</math>. In the other direction, the condition is sufficient, as can be shown by induction. 
More strongly, if the factorization of <math>n</math> satisfies the condition above, then any <math>m \le \sigma(n)</math> can be represented as a sum of divisors of <math>n</math>, by the following sequence of steps:<ref>{{harvtxt|Stewart|1954}}; {{harvtxt|Sierpiński|1955}}.</ref>
* By induction on <math>j\in[1,\alpha_k]</math>, it can be shown that <math>p_k^j\leq 1+\sigma(n/p_k^{\alpha_k-(j-1)})</math>. Hence <math>p_k^{\alpha_k}\leq 1+\sigma(n/p_k)</math>.
* Since the internals <math>[q p_k^{\alpha_k}, q p_k^{\alpha_k}+\sigma(n/p_k)]</math> cover <math>[1,\sigma(n)]</math> for <math>1\leq q\leq \sigma(n/p_k^{\alpha_k})</math>, there are such a <math>q</math> and some <math>r\in[0,\sigma(n/p_k)]</math> such that <math>m=q p_k^{\alpha_k}+r</math>.
* Since <math>q\le\sigma(n/p_k^{\alpha_k})</math> and <math>n/p_k^{\alpha_k}</math> can be shown by induction to be practical, we can find a representation of ''q'' as a sum of divisors of <math>n/p_k^{\alpha_k}</math>.
* Since <math>r\le \sigma(n/p_k)</math>, and since <math>n/p_k</math> can be shown by induction to be practical, we can find a representation of ''r'' as a sum of divisors of <math>n/p_k</math>.
* The divisors representing ''r'', together with <math>p_k^{\alpha_k}</math> times each of the divisors representing ''q'', together form a representation of ''m'' as a sum of divisors of <math>n</math>.