Definition
A positive integer is called B-smooth if none of its prime factors is greater than B. For example, 1,620 has prime factorization 22 × 34 × 5; therefore 1,620 is 5-smooth because none of its prime factors is greater than 5. It is worth note that this definition includes numbers that lack some of the smaller prime factors. For example, both 10 and 12 are 5-smooth, despite the fact that they miss out prime factors 3 and 5 respectively. 5-smooth numbers are also called regular numbers or Hamming numbers; 7-smooth numbers are sometimes called highly composite or humble, although this conflicts with another meaning of that term.
Note that B does not have to be a prime factor. If the largest prime factor of a number is p then the number is B-smooth for any B ≥ p. Usually B is given as a prime, but composite numbers work as well. A number is B-smooth if and only if it is p-smooth, where p is the largest prime less than or equal to B.
Read more about this topic: Smooth Number
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