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:
“Science is the language of the temporal world; love is that of the spiritual world. Man, indeed, describes more than he explains; while the angelic spirit sees and understands. Science saddens man; love enraptures the angel; science is still seeking, love has found. Man judges of nature in relation to itself; the angelic spirit judges of it in relation to heaven. In short to the spirits everything speaks.”
—Honoré De Balzac (1799–1850)
“We shall never resolve the enigma of the relation between the negative foundations of greatness and that greatness itself.”
—Jean Baudrillard (b. 1929)
“There are three classes into which all the women past seventy that ever I knew were to be divided: 1. That dear old soul; 2. That old woman; 3. That old witch.”
—Samuel Taylor Coleridge (1772–1834)
“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)