practicalnumbersdisiorrn

A practical number is a positive integer n such that all smaller positive integers can be represented as sums of distinct divisors of n. For example, the divisors of 6 are 1, 2, and 3. The integers less than 6 are 1, 2, 3, 4, and 5. These can be represented as sums of distinct divisors of 6:

1 = 1

2 = 2

3 = 3

4 = 1 + 3

5 = 2 + 3

Therefore, 6 is a practical number.

The sequence of practical numbers begins 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, ...

A characterization of practical numbers was provided by Stewart in 1954. A positive integer n with prime factorization $n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$ where $p_1 < p_2 < \cdots < p_k$ are prime numbers and $a_i \ge 1$ is a practical number if and only if $p_1 = 2$ and for each $i$ from 2 to $k$, the inequality $p_i \le \sigma(p_1^{a_1} p_2^{a_2} \cdots p_{i-1}^{a_{i-1}}) + 1$ holds, where $\sigma(m)$ is the sum of the divisors of m.

Every power of 2 is a practical number. Also, if n is a practical number, then $n \cdot 2^k$ is a practical number for any integer $k \ge 0$. Practical numbers are related to the concept of abundant numbers and perfect numbers. Every perfect number is a practical number.

The set of practical numbers is closed under multiplication. If m and n are practical numbers, then their product mn is also a practical number. This property simplifies the generation and identification of practical numbers.

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