aka “Hamming numbers”, “ugly numbers”.

Prime factors of those numbers are 2, 3 or 5.

How to implement?

• 1 is the first ugly number.
• If R is an ugly number, then 2R, 3R and 5*R are all ugly numbers as well.
• There are many ugly numbers so we need a priority queue to select them in ascending order.
• Suppose R equals to 2^a * 3^b, should 2R be pushed into the priority queue? Considering another number R’ which equals to 2^(a+1) * 3^(b-1), we known that R’ < R. So even if both of R’ & R are in the priority queue, R’ would be selected before R. When R’ is selected, 3R’ should be pushed into priority queue at the same time. Which means 3R’ is pushed into queue before 2R, so we do not have to push 2R again (3R’ = 2^(a+1) * 3^b = 2*R).
• So every time when a number R is selected, new ugly number should be generated by multiply the selected number with a new factor P, and P must greater or equal to the largest prime factor of R.
• To illustrate the idea, build a tree with root node 1 (the first ugly number). All node values in this tree are their parent values multiple 2, 3, or 5 (e.g. children of 1 are 2, 3 and 5. children of 2 are 22, 23 and 2*5. etc.). Remove the duplicated numbers and selected remains in ascending order.