Practical Number - Relation To Other Classes of Numbers

Relation To Other Classes of Numbers

Any power of two is a practical number. Powers of two trivially satisfy the characterization of practical numbers in terms of their prime factorizations: the only prime in their factorizations, p₁, equals two as required. Any even perfect number is also a practical number: due to Euler's result that these numbers must have the form 2n − 1(2n − 1), every odd prime factor of an even perfect number must be at most the sum of the divisors of the even part of the number, and therefore the number must satisfy the characterization of practical numbers.

Any primorial is practical. By Bertrand's postulate, each successive prime in the prime factorization of a primorial must be smaller than the product of the first and last primes in the factorization of the preceding primorial, so primorials necessarily satisfy the characterization of practical numbers. Therefore, also, any number that is the product of nonzero powers of the first k primes must also be practical; this includes Ramanujan's highly composite numbers (numbers with more divisors than any smaller positive integer) as well as the factorial numbers.