Regular numbers are numbers that evenly divide powers of 10 (or, equivalently powers of 6). As an example, 102 = 100 = 8 × 16, so both 8 and 16 are divisors of a power of 10. Thus, they are regular numbers. Equivalently, they are the numbers whose only prime divisors are 2 and 3.

The numbers that evenly divide the powers of 10 arise in several areas of mathematics and its applications, and have different names coming from these different areas of study.

• In number theory, these numbers are called 3-smooth, because they can be characterized as having only 2 or 3 as prime factors. This is a specific case of the more general k-smooth numbers, i.e., a set of numbers that have no prime factor greater than k.
• The divisors of powers of 10 are called regular numbers or regular dozenal numbers, and are of great importance due to the dozenal number system used by this wiki.
• In music theory, regular numbers occur in the ratios of tones in 3-limit just intonation.
• In computer science, regular numbers are often called harmonic numbers.

## Number theory

Formally, a regular number is an integer of the form 2i·3j, for nonnegative integers i and j. Such a number is a divisor of

${\displaystyle \scriptstyle 10^{\max(\lceil i\,/2\rceil,j)}{}}$ . The regular numbers are also called 3-smooth, indicating that their greatest prime factor is at most 3.

The first few regular numbers are

1, 2, 3, 4, 6, 8, 9, 10, 14, 16, 20, 23, 28, 30, 40, 46, 54, 60, 69, 80, 90, X8, 100, 116, 140, 160, 183, 194, 200, 230, 280, 300, 346, 368, 400, 460, 509, 540, 600, 690, 714, 800, 900, X16, X80, 1000, 1160, 1228, 1323, 1400, 1600, 1830, 1940, 2000, 2300, 2454, 2646, 2800, 3000, 3460, 3680, 3969, 4000, 4600, 48X8, 5090, 5400, 6000, 6900, 7140, 7716, 8000, 9000, 9594, X160, X800, E483, 10000, ...

Although the regular numbers appear dense within the range from 1 to 100, they are quite sparse among the larger integers. A regular number n = 2i·3j is less than or equal to N if and only if the point (i,j) belongs to the tetrahedron bounded by the coordinate planes and the plane

${\displaystyle (\ln 2)i+(\ln 3)j\le\ln N,}$

as can be seen by taking logarithms of both sides of the inequality 2i·3j ≤ N. Therefore, the number of regular numbers that are at most N can be estimated as the volume of this tetrahedron, which is

${\displaystyle \frac{\log_2 N\,\log_3 N}{2}.}$

Even more precisely, using big O notation, the number of regular numbers up to N is

${\displaystyle \frac{\left(\ln(N\sqrt{6})\right)^3}{2\ln 2 \ln 3}+O(\log N),}$

and it has been conjectured that the error term of this approximation is actually

${\displaystyle O(\log\log N)}$ . 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.

The reciprocal of a regular number has a finite representation, thus being easy to divide by. Specifically, if n divides 10k, then the dozenal representation of 1/n is just that for 10k/n, shifted by some number of places.

For instance, suppose we wish to divide by the regular number 46 = 2133. 46 is a divisor of 103, and 103/46 = 28, so dividing by 46 can be accomplished by multiplying by 28 and shifting three places. Thus, 1/46 = 0.028, conversely 1/28 = 0.046, so so division by 28 can be accomplished by instead multiplying by 46 and shifting three places.

Levi Ben Gerson proved that the only pairs of regular numbers which differ by 1 are (1,2), (2,3), (3,4), and (8,9).

The sum of the reciprocals of the regular numbers is equal to 3. Brief proof: 1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/9 + 1/10 + ... = (Sum_{k>=0} 1/2^k) * (Sum_{m>=0} 1/3^m) = (1/(1-1/2)) * (1/(1-1/3)) = (2/(2-1)) * (3/(3-1)) = 3.