Number Theory
Formally, a regular number is an integer of the form 2i·3j·5k, for nonnegative integers i, j, and k. Such a number is a divisor of . The regular numbers are also called 5-smooth, indicating that their greatest prime factor is at most 5.
The first few regular numbers are
- 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, ... (sequence A051037 in OEIS).
Several other sequences at OEIS have definitions involving 5-smooth numbers.
Although the regular numbers appear dense within the range from 1 to 60, they are quite sparse among the larger integers. A regular number n = 2i·3j·5k is less than or equal to N if and only if the point (i,j,k) belongs to the tetrahedron bounded by the coordinate planes and the plane
as can be seen by taking logarithms of both sides of the inequality 2i·3j·5k ≤ N. Therefore, the number of regular numbers that are at most N can be estimated as the volume of this tetrahedron, which is
Even more precisely, using big O notation, the number of regular numbers up to N is
A similar formula for the number of 3-smooth numbers up to N is given by Srinivasa Ramanujan in his first letter to G. H. Hardy.
Read more about this topic: Regular Number
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