regulaararved

Regulaararved are positive integers whose prime factors do not exceed 5. In other words, a regulaararv can be written as 2^a 3^b 5^c with nonnegative integers a, b, and c. They are commonly known in English literature as 5-smooth numbers or Hamming numbers.

Examples of regulaararved start 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, 60, 64, 72, 75, 80, 90, 96, 100, 108, 120, 125, 128, 135, 144, 150, 160, 162, 180, 192, 200, and so on. Every regulaararv can be formed by multiplying powers of 2, 3, and 5, and conversely, any such product is a regulaararv.

Properties and implications: The set of regulaararved is closed under multiplication, since multiplying two numbers with prime factors only in {2,3,5} yields another such number. They form a monotonically increasing sequence with gaps that grow as numbers increase. The density of regulaararved up to a threshold x is low; asymptotically, their count up to x is about (log x)^3 divided by (6 log 2 log 3 log 5). They can be generated in increasing order by merging three sequences obtained by multiplying by 2, 3, or 5, or by using a priority queue approach; this is a common technique in programming problems involving “Ugly Numbers.”

See also: 5-smooth numbers, Hamming numbers, Ugly numbers.