3smooth

3smooth refers to the positive integers whose prime factors are only 2 and 3. Equivalently, every 3smooth number can be written in the form 2^a 3^b with a and b nonnegative integers. They are also called 2-3-smooth numbers or, in some contexts, simply 3-smooth numbers. This makes them a simple case of y-smooth or p-smooth numbers studied in number theory.

The sequence begins with 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192, 216, 243, 256, 288, 324, 384, 432, 486, 512, 576, 648, 729, 768, 864, 972, 1024, and so on. Each member is obtained by multiplying any previous 3smooth number by 2 or by 3, preserving the restriction on prime factors. The set is closed under multiplication by 2 or 3, and 1 serves as the multiplicative identity.

Counting these numbers up to a bound x has a known asymptotic: the number N_3(x) of 3smooth integers not exceeding x satisfies N_3(x) ~ (log x)^2 / (2 log 2 log 3) as x grows large. This reflects the two degrees of freedom in the exponents a and b in n = 2^a 3^b.

3smooth numbers appear in problems that prefer small prime factors, in certain algorithmic and combinatorial contexts, and as a convenient test set in multiplicative sequence constructions. They are a fundamental example in the broader study of smooth numbers.