practical number and is one of its divisors then must also be a practical number. In the set of all practical numbers there is a primitive set of practical numbers. A primitive practical number is either practical and squarefree or practical and when divided by any of its prime factors whose factorization exponent is greater than 1 is no longer practical. The sequence of primitive practical numbers begins Every positive integer has a practical multiple. For instance, for every integer , its multiple is practical. Relation to other classes of numbers Several other notable sets of integers consist only of practical numbers: From the above properties with a practical number and one of its divisors (that is, ) then must also be a practical number therefore six times every power of 3 must be a practical number as well as six times every power of 2. Every 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. Every even perfect number is also a practical number. This follows from Leonhard Euler's result that an even perfect number must have the form . The odd part of this factorization equals the sum of the divisors of the even part, so every odd prime factor of such a number must be at most the sum of the divisors of the even part of the number. Therefore, this number must satisfy the characterization of practical numbers. Every primorial (the product of the first primes, for some ) is practical. For the first two primorials, two and six, this is clear. Each successive primorial is formed by multiplying a prime number by a smaller primorial that is divisible by both two and the next smaller prime, . By Bertrand's postulate, , so each successive prime factor in the primorial is less than one of the divisors of the previous primorial. By induction, it follows that every primorial satisfies the characterization of practical numbers. Because a primorial is, by definition, squarefree it is also a primitive practical number. Generalizing the primorials, any number that is the product of nonzero powers of the first 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. Practical numbers