5smooth

5smooth numbers, also called 5-smooth numbers, are positive integers whose prime factors are only 2, 3, and 5. Equivalently, they are numbers that can be written in the form 2^a 3^b 5^c with nonnegative integers a, b, and c. They are a subset of the broader class of smooth numbers and are sometimes referred to as Hamming numbers.

The sequence begins as 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, and so on. Because multiplying two 5-smooth numbers yields another 5-smooth number, the set is closed under multiplication and forms an infinite, well-ordered sequence. They occur frequently in computational problems and have practical uses in areas such as algorithm design and numerical methods.

A common way to generate 5-smooth numbers in increasing order is the three-pointer method. Start with the list containing 1, and maintain indices for multiplying by 2, 3, and 5. Repeatedly append the smallest candidate among 2 times the 2-pointer, 3 times the 3-pointer, and 5 times the 5-pointer, then advance all pointers that produced the appended value. This approach ensures duplicates are avoided and the sequence remains sorted.

In mathematics, 5-smooth numbers illustrate the broader concept of smooth numbers (numbers with restricted prime factors) and relate to problems in number theory, combinatorics, and computer science.