Regular number

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

These numbers arise in several areas of mathematics and its applications, and have different names coming from their different areas of study.

• In
  prime factors. This is a specific case of the more general k-smooth numbers
  , the numbers that have no prime factor greater than k.
• In the study of Babylonian mathematics, the divisors of powers of 60 are called regular numbers or regular sexagesimal numbers, and are of great importance in this area because of the sexagesimal (base 60) number system that the Babylonians used for writing their numbers, and that was central to Babylonian mathematics.
• In music theory, regular numbers occur in the ratios of tones in five-limit just intonation. In connection with music theory and related theories of architecture, these numbers have been called the harmonic whole numbers.
• In computer science, regular numbers are often called Hamming numbers, after Richard Hamming, who proposed the problem of finding computer algorithms for generating these numbers in ascending order. This problem has been used as a test case for functional programming.

Number theory

Formally, a regular number is an integer of the form ${\displaystyle 2^{i}\cdot 3^{j}\cdot 5^{k}}$, for nonnegative integers ${\displaystyle i}$, ${\displaystyle j}$, and ${\displaystyle k}$. Such a number is a divisor of ${\displaystyle 60^{\max(\lceil i\,/2\rceil ,j,k)}}$. The regular numbers are also called 5-

The first few regular numbers are[2]

Several other sequences at the On-Line Encyclopedia of Integer Sequences have definitions involving 5-smooth numbers.^[4]

Although the regular numbers appear dense within the range from 1 to 60, they are quite sparse among the larger integers. A regular number ${\displaystyle n=2^{i}\cdot 3^{j}\cdot 5^{k}}$ is less than or equal to some threshold ${\displaystyle N}$ if and only if the point ${\displaystyle (i,j,k)}$ belongs to the tetrahedron bounded by the coordinate planes and the plane

$${\displaystyle i\ln 2+j\ln 3+k\ln 5\leq \ln N,}$$

${\displaystyle 2^{i}\cdot 3^{j}\cdot 5^{k}\leq N}$

${\displaystyle N}$

$${\displaystyle {\frac {\log _{2}N\,\log _{3}N\,\log _{5}N}{6}}.}$$

${\displaystyle N}$

$${\displaystyle {\frac {\left(\ln(N{\sqrt {30}})\right)^{3}}{6\ln 2\ln 3\ln 5}}+O(\log N),}$$

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

${\displaystyle N}$

Babylonian mathematics

In the Babylonian sexagesimal notation, the reciprocal of a regular number has a finite representation. If ${\displaystyle n}$ divides ${\displaystyle 60^{k}}$, then the sexagesimal representation of ${\displaystyle 1/n}$ is just that for ${\displaystyle 60^{k}/n}$, shifted by some number of places. This allows for easy division by these numbers: to divide by ${\displaystyle n}$, multiply by ${\displaystyle 1/n}$, then shift.^[6]

For instance, consider division by the regular number 54 = 2¹3³. 54 is a divisor of 60³, and 60³/54 = 4000, so dividing by 54 in sexagesimal can be accomplished by multiplying by 4000 and shifting three places. In sexagesimal 4000 = 1×3600 + 6×60 + 40×1, or (as listed by Joyce) 1:6:40. Thus, 1/54, in sexagesimal, is 1/60 + 6/60² + 40/60³, also denoted 1:6:40 as Babylonian notational conventions did not specify the power of the starting digit. Conversely 1/4000 = 54/60³, so division by 1:6:40 = 4000 can be accomplished by instead multiplying by 54 and shifting three sexagesimal places.

The Babylonians used tables of reciprocals of regular numbers, some of which still survive.

Although the primary reason for preferring regular numbers to other numbers involves the finiteness of their reciprocals, some Babylonian calculations other than reciprocals also involved regular numbers. For instance, tables of regular squares have been found[6] and the broken tablet Plimpton 322 has been interpreted by Neugebauer as listing Pythagorean triples ${\displaystyle (p^{2}-q^{2},\,2pq,\,p^{2}+q^{2})}$ generated by ${\displaystyle p}$ and ${\displaystyle q}$ both regular and less than 60.^[10] Fowler and Robson discuss the calculation of square roots, such as how the Babylonians found an approximation to the square root of 2, perhaps using regular number approximations of fractions such as 17/12.^[9]

Music theory

In music theory, the just intonation of the diatonic scale involves regular numbers: the pitches in a single octave of this scale have frequencies proportional to the numbers in the sequence 24, 27, 30, 32, 36, 40, 45, 48 of nearly consecutive regular numbers.^[11] Thus, for an instrument with this tuning, all pitches are regular-number harmonics of a single fundamental frequency. This scale is called a 5-limit tuning, meaning that the interval between any two pitches can be described as a product 2ⁱ3^j5^k of powers of the prime numbers up to 5, or equivalently as a ratio of regular numbers.^[12]

5-limit musical scales other than the familiar diatonic scale of Western music have also been used, both in traditional musics of other cultures and in modern experimental music:

In connection with the application of regular numbers to music theory, it is of interest to find pairs of regular numbers that differ by one. There are exactly ten such pairs ${\displaystyle (x,x+1)}$ and each such pair defines a

superparticular ratio

${\displaystyle {\tfrac {x+1}{x}}}$ that is meaningful as a musical interval. These intervals are 2/1 (the

• The sequence of Hamming numbers begins with the number 1.
• The remaining values in the sequence are of the form ${\displaystyle 2h}$, ${\displaystyle 3h}$, and ${\displaystyle 5h}$, where ${\displaystyle h}$ is any Hamming number.
• Therefore, the sequence ${\displaystyle H}$ may be generated by outputting the value 1, and then merging the sequences ${\displaystyle 2H}$, ${\displaystyle 3H}$, and ${\displaystyle 5H}$.

This algorithm is often used to demonstrate the power of a

In the Python programming language, lazy functional code for generating regular numbers is used as one of the built-in tests for correctness of the language's implementation.^[19]

A related problem, discussed by Knuth (1972), is to list all ${\displaystyle k}$-digit sexagesimal numbers in ascending order (see #Babylonian mathematics above). In algorithmic terms, this is equivalent to generating (in order) the subsequence of the infinite sequence of regular numbers, ranging from ${\displaystyle 60^{k}}$ to ${\displaystyle 60^{k+1}}$.^[8] See Gingerich (1965) for an early description of computer code that generates these numbers out of order and then sorts them;^[20] Knuth describes an ad hoc algorithm, which he attributes to Bruins (1970), for generating the six-digit numbers more quickly but that does not generalize in a straightforward way to larger values of ${\displaystyle k}$.^[8] Eppstein (2007) describes an algorithm for computing tables of this type in linear time for arbitrary values of ${\displaystyle k}$.^[21]

Other applications

Heninger, Rains & Sloane (2006) show that, when ${\displaystyle n}$ is a regular number and is divisible by 8, the generating function of an ${\displaystyle n}$-dimensional extremal even unimodular lattice is an ${\displaystyle n}$th power of a polynomial.^[22]

As with other classes of

Certain species of bamboo release large numbers of seeds in synchrony (a process called masting) at intervals that have been estimated as regular numbers of years, with different intervals for different species, including examples with intervals of 10, 15, 16, 30, 32, 48, 60, and 120 years.^[25] It has been hypothesized that the biological mechanism for timing and synchronizing this process lends itself to smooth numbers, and in particular in this case to 5-smooth numbers. Although the estimated masting intervals for some other species of bamboo are not regular numbers of years, this may be explainable as measurement error.^[25]

Notes

References