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.
Read more about this topic: Practical Number
Famous quotes containing the words relation to, relation, classes and/or numbers:
“Any relation to the land, the habit of tilling it, or mining it, or even hunting on it, generates the feeling of patriotism. He who keeps shop on it, or he who merely uses it as a support to his desk and ledger, or to his manufactory, values it less.”
—Ralph Waldo Emerson (1803–1882)
“There is a relation between the hours of our life and the centuries of time. As the air I breathe is drawn from the great repositories of nature, as the light on my book is yielded by a star a hundred millions of miles distant, as the poise of my body depends on the equilibrium of centrifugal and centripetal forces, so the hours should be instructed by the ages and the ages explained by the hours.”
—Ralph Waldo Emerson (1803–1882)
“Classes struggle, some classes triumph, others are eliminated. Such is history; such is the history of civilization for thousands of years.”
—Mao Zedong (1893–1976)
“I had a feeling that out there, there were very poor people who didn’t have enough to eat. But they wore wonderfully colored rags and did musical numbers up and down the streets together.”
—Jill Robinson (b. 1936)