Regular number

Regular numbers are numbers that evenly divide powers of 60 (or, equivalently powers of 30). As an example, 60² = 3600 = 48 × 75, so both 48 and 75 are divisors of a power of 60. Thus, they are regular numbers. Equivalently, they are the numbers whose only prime divisors are 2, 3, and 5.

The numbers that evenly divide the powers of 60 arise in several areas of mathematics and its applications, and have different names coming from these different areas of study.

Formally, a regular number is an integer of the form 2ⁱ·3^j·5^k, for nonnegative integers i, j, and k. Such a number is a divisor of ${\displaystyle \scriptstyle 60^{\max(\lceil i\,/2\rceil ,j,k)}{}}$. The regular numbers are also called 5-smooth, indicating that their greatest prime factor is at most 5.

The first few regular numbers are

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·3^j·5^k 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·3^j·5^k ≤ N. Therefore, the number of regular numbers that are at most N can be estimated as the volume of this tetrahedron, which is

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