Editing Smooth number (section)

==Definition==
A [[negative and positive numbers|positive]] [[integer]] is called <var>B</var>-'''smooth''' if none of its [[prime factor]]s are greater than <var>B</var>. For example, 1,620 has prime factorization 2<sup>2</sup> × 3<sup>4</sup> × 5; therefore 1,620 is 5-smooth because none of its prime factors are greater than 5. This definition includes numbers that lack some of the smaller prime factors; for example, both 10 and 12 are 5-smooth, even though they miss out the prime factors 3 and 5, respectively. All 5-smooth numbers are of the form 2<sup>''a''</sup> × 3<sup>''b''</sup> × 5<sup>''c''</sup>, where ''a'', ''b'' and ''c'' are non-negative integers.

The 3-smooth numbers have also been called "harmonic numbers", although that name has other more widely used meanings.<ref>{{cite OEIS|A003586|3-smooth numbers}}</ref>
5-smooth numbers are also called [[regular number|'''regular numbers''']] or Hamming numbers;<ref>{{Cite web|url=https://www.w3resource.com/python-exercises/challenges/1/python-challenges-1-exercise-25.php|title=Python: Get the Hamming numbers upto a given numbers also check whether a given number is an Hamming number|website=w3resource|language=en|access-date=2019-12-12}}</ref> 7-smooth numbers are also called '''humble numbers''',<ref>{{Cite web|url=https://www.eecs.qmul.ac.uk/~pbo/ACM/archive/00001.html|title=Problem H: Humble Numbers|website=www.eecs.qmul.ac.uk|access-date=2019-12-12}}</ref> and sometimes called ''highly composite'',<ref>{{Cite OEIS|A002473|name=7-smooth numbers}}</ref> although this conflicts with another meaning of [[highly composite numbers]].

Here, note that <var>B</var> itself is not required to appear among the factors of a <var>B</var>-smooth number. If the largest prime factor of a number is <var>p</var> then the number is <var>B</var>-smooth for any <var>B</var> ≥ <var>p</var>. In many scenarios <var>B</var> is [[Prime number|prime]], but [[composite number]]s are permitted as well. A number is <var>B</var>-smooth [[if and only if]] it is <var>p</var>-smooth, where <var>p</var> is the largest prime less than or equal to <var>B</var>.