View source for Regular number

{{good article}}
{{short description|Numbers that evenly divide powers of 60}}
{{distinguish|regular prime}}
[[File:Regular divisibility lattice.svg|thumb|upright=1.35|A [[Hasse diagram]] of [[divisibility]] relationships among the regular numbers up to 400. The vertical scale is [[Logarithmic scale|logarithmic]].<ref>Inspired by similar diagrams by Erkki Kurenniemi in [http://www.beige.org/projects/dimi/CSDL2.pdf "Chords, scales, and divisor lattices"].</ref>]]

'''Regular numbers''' are numbers that evenly divide powers of [[60 (number)|60]] (or, equivalently, powers of [[30 (number)|30]]).  Equivalently, they are the numbers whose only prime divisors are [[2]], [[3]], and [[5]]. As an example, 60<sup>2</sup>&nbsp;=&nbsp;3600&nbsp;=&nbsp;48&nbsp;&times;&nbsp;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 [[number theory]], these numbers are called '''5-smooth''', because they can be characterized as having only 2, 3, or 5 as their [[prime factor]]s. This is a specific case of the more general {{mvar|k}}-[[smooth number]]s, the numbers that have no prime factor greater {{nowrap|than {{mvar|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 tuning|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 [[algorithm]]s 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 <math>2^i\cdot 3^j\cdot 5^k</math>, for nonnegative integers <math>i</math>, <math>j</math>, and <math>k</math>. Such a number is a divisor of <math>60^{\max(\lceil i\,/2\rceil,j,k)}</math>. The regular numbers are also called 5-[[smooth number|smooth]], indicating that their greatest [[prime factor]] is at most 5.{{sfnp|Sloane "A051037"}} More generally, a {{mvar|k}}-smooth number is a number whose greatest prime factor is at {{nowrap|most {{mvar|k}}.{{sfnp|Pomerance|1995}}}}

The first few regular numbers are{{sfnp|Sloane "A051037"}}
{{block indent|left=1.6|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, ... {{OEIS|A051037}}}}
Several other sequences at the [[On-Line Encyclopedia of Integer Sequences]] have definitions involving 5-smooth numbers.<ref>[http://www.oeis.org/search?q=5-smooth OEIS search for sequences involving 5-smoothness].</ref>

Although the regular numbers appear dense within the range from 1 to 60, they are quite sparse among the larger integers. A regular number <math>n=2^i\cdot 3^j\cdot 5^k</math> is less than or equal to some threshold <math>N</math> if and only if the point <math>(i,j,k)</math> belongs to the [[tetrahedron]] bounded by the coordinate planes and the plane
<math display=block>i\ln 2+j\ln 3+k\ln 5\le\ln N,</math>
as can be seen by taking logarithms of both sides of the inequality <math>2^i\cdot 3^j\cdot 5^k\le N</math>.
Therefore,  the number of regular numbers that are at most <math>N</math> can be estimated as the [[volume]] of this tetrahedron, which is
<math display=block>\frac{\log_2 N\,\log_3 N\,\log_5 N}{6}.</math>
Even more precisely, using [[big O notation]], the number of regular numbers up to <math>N</math> is
<math display=block>\frac{\left(\ln(N\sqrt{30})\right)^3}{6\ln 2 \ln 3 \ln 5}+O(\log N),</math>
and it has been conjectured that the error term of this approximation is actually <math>O(\log\log N)</math>.{{sfnp|Sloane "A051037"}}
A similar formula for the number of 3-smooth numbers up to <math>N</math> is given by [[Srinivasa Ramanujan]] in his first letter to [[G. H. Hardy]].{{sfnp|Berndt|Rankin|1995}}

== Babylonian mathematics ==
[[File:Table de division et de conversion des fractions - Louvre - AO 6456.JPG|thumb|upright=1.35|AO 6456, a table of reciprocals of regular numbers from [[Uruk#Uruk into Late Antiquity|Seleucid Uruk]], copied from an unknown earlier source]]
In the Babylonian [[sexagesimal]] notation, the [[Multiplicative inverse|reciprocal]] of a regular number has a finite representation. If <math>n</math> divides <math>60^k</math>, then the sexagesimal representation of <math>1/n</math> is just that for <math>60^k/n</math>, shifted by some number of places. This allows for easy division by these numbers: to divide by <math>n</math>, multiply by <math>1/n</math>, then shift.{{sfnp|Aaboe|1965}}

For instance, consider division by the regular number 54 = 2<sup>1</sup>3<sup>3</sup>. 54 is a divisor of 60<sup>3</sup>, and 60<sup>3</sup>/54 = 4000, so dividing by 54 in sexagesimal can be accomplished by multiplying by 4000 and shifting three places. In sexagesimal 4000 = 1&times;3600 + 6&times;60 + 40&times;1, or (as listed by Joyce) 1:6:40. Thus, 1/54, in sexagesimal, is 1/60 + 6/60<sup>2</sup> + 40/60<sup>3</sup>, also denoted 1:6:40 as Babylonian notational conventions did not specify the power of the starting digit. Conversely 1/4000 = 54/60<sup>3</sup>, 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.{{sfnp|Sachs|1947}} These tables existed relatively unchanged throughout Babylonian times.{{sfnp|Aaboe|1965}} One tablet from [[Seleucid]] times, by someone named Inaqibıt-Anu, contains the reciprocals of 136 of the 231 six-place regular numbers whose first place is 1 or 2, listed in order. It also includes reciprocals of some numbers of more than six places, such as 3{{sup|23}} (2 1 4 8 3 0 7 in sexagesimal), whose reciprocal has 17 sexagesimal digits. Noting the difficulty of both calculating these numbers and sorting them, [[Donald Knuth]] in 1972 hailed Inaqibıt-Anu as "the first man in history to solve a computational problem that takes longer than one second of time on a modern electronic computer!" (Two tables are also known giving approximations of reciprocals of non-regular numbers, one of which gives reciprocals for all the numbers from 56 to 80.){{sfnp|Knuth|1972}}{{sfnp|Fowler|Robson|1998}}

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{{sfnp|Aaboe|1965}} and the broken tablet [[Plimpton 322]] has been  interpreted by [[Otto E. Neugebauer|Neugebauer]] as listing [[Pythagorean triple]]s <math>( p^2 - q^2,\, 2pq,\, p^2 + q^2 )</math> generated by <math>p</math> and <math>q</math> both regular and less than 60.<ref>See {{harvtxt|Conway|Guy|1996}} for a popular treatment of this interpretation. [[Plimpton 322]] has other interpretations, for which see its article, but all involve regular numbers.</ref>
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.{{sfnp|Fowler|Robson|1998}}

== Music theory ==

In [[music theory]], the [[just intonation]] of the [[diatonic scale]] involves regular numbers: the [[pitch (music)|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.{{sfnp|Clarke|1877}} Thus, for an instrument with this tuning, all pitches are regular-number [[harmonic]]s of a single [[fundamental frequency]]. This scale is called a 5-[[limit (music)|limit]] tuning, meaning that the [[interval (music)|interval]] between any two pitches can be described as a product 2<sup>i</sup>3<sup>j</sup>5<sup>k</sup> of powers of the prime numbers up to 5, or equivalently as a ratio of regular numbers.{{sfnp|Honingh|Bod|2005}}

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: {{harvtxt|Honingh|Bod|2005}} list 31 different 5-limit scales, drawn from a larger database of musical scales. Each of these 31 scales shares with diatonic just intonation the property that all intervals are ratios of regular numbers.{{sfnp|Honingh|Bod|2005}} [[Euler]]'s [[tonnetz]] provides a convenient graphical representation of the pitches in any 5-limit tuning, by factoring out the octave relationships (powers of two) so that the remaining values form a planar [[Regular grid|grid]].{{sfnp|Honingh|Bod|2005}} Some music theorists have stated more generally that regular numbers are fundamental to tonal music itself, and that pitch ratios based on primes larger than 5 cannot be [[Consonance and dissonance|consonant]].<ref>{{harvtxt|Asmussen|2001}}, for instance, states that "within any piece of tonal music" all intervals must be ratios of regular numbers, echoing similar statements by much earlier writers such as {{harvtxt|Habens|1889}}. In the modern music theory literature this assertion is often attributed to {{harvtxt|Longuet-Higgins|1962}}, who used a graphical arrangement closely related to the [[tonnetz]] to organize 5-limit pitches.</ref> However the [[equal temperament]] of modern pianos is not a 5-limit tuning,{{sfnp|Kopiez|2003}} and some modern composers have experimented with tunings based on primes larger than five.{{sfnp|Wolf|2003}}

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 <math>(x,x+1)</math> and each such pair defines a [[Superparticular number|superparticular ratio]] <math>\tfrac{x+1}{x}</math> that is meaningful as a musical interval. These intervals are 2/1 (the [[octave]]), 3/2 (the [[perfect fifth]]), 4/3 (the [[perfect fourth]]), 5/4 (the [[just major third]]), 6/5 (the [[just minor third]]), 9/8 (the [[just major tone]]), 10/9 (the [[just minor tone]]), 16/15 (the [[just diatonic semitone]]), 25/24 (the [[just chromatic semitone]]), and 81/80 (the [[syntonic comma]]).<ref>{{harvtxt|Halsey|Hewitt|1972}} note that this follows from [[Størmer's theorem]] {{harv|Størmer|1897}}, and provide a proof for this case; see also {{harvtxt|Silver|1971}}.</ref>

In the Renaissance theory of [[Musica universalis|universal harmony]], musical ratios were used in other applications, including the [[architecture]] of buildings. In connection with the analysis of these shared musical and architectural ratios, for instance in the architecture of [[Palladio]], the regular numbers have also been called the '''harmonic whole numbers'''.{{sfnp|Howard|Longair|1982}}

==Algorithms==
{{Wikifunctions|Z15224|regular number checking}}
Algorithms for calculating the regular numbers in ascending order were popularized by [[Edsger Dijkstra]]. {{harvs|last=Dijkstra||year=1976|year2=1981|txt}} attributes to Hamming the problem of building the infinite ascending sequence of all 5-smooth numbers; this problem is now known as '''Hamming's problem''', and the numbers so generated are also called the '''Hamming numbers'''. Dijkstra's ideas to compute these numbers are the following:

* The sequence of Hamming numbers begins with the number 1.
* The remaining values in the sequence are of the form <math>2h</math>, <math>3h</math>, and <math>5h</math>, where <math>h</math> is any Hamming number.
* Therefore, the sequence <math>H</math> may be generated by outputting the value 1, and then [[Merge algorithm|merging]] the sequences <math>2H</math>, <math>3H</math>, and <math>5H</math>.

This algorithm is often used to demonstrate the power of a [[Lazy evaluation|lazy]] [[functional programming language]], because (implicitly) concurrent efficient implementations, using a constant number of arithmetic operations per generated value, are easily constructed as described above. Similarly efficient strict functional or [[imperative programming language|imperative]] sequential implementations are also possible whereas explicitly concurrent [[generator (computer programming)|generative]] solutions might be non-trivial.<ref>See, e.g., {{harvtxt|Hemmendinger|1988}} or {{harvtxt|Yuen|1992}}.</ref>

In the [[Python (programming language)|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.<ref>Function m235 in [http://svn.python.org/projects/python/trunk/Lib/test/test_generators.py test_generators.py].</ref>

A related problem, discussed by {{harvtxt|Knuth|1972}}, is to list all <math>k</math>-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 <math>60^k</math> to <math>60^{k+1}</math>.{{sfnp|Knuth|1972}}
See {{harvtxt|Gingerich|1965}} for an early description of computer code that generates these numbers out of order and then sorts them;{{sfnp|Gingerich|1965}} Knuth describes an ad hoc algorithm, which he attributes to {{harvtxt|Bruins|1970}}, for generating the six-digit numbers more quickly but that does not generalize in a straightforward way to larger values of <math>k</math>.{{sfnp|Knuth|1972}}  {{Harvtxt|Eppstein|2007}} describes an algorithm for computing tables of this type in linear time for arbitrary values of <math>k</math>.{{sfnp|Eppstein|2007}}

== Other applications ==
{{harvtxt|Heninger|Rains|Sloane|2006}} show that, when <math>n</math> is a regular number and is divisible by 8, the generating function of an <math>n</math>-dimensional extremal even [[unimodular lattice]] is an <math>n</math>th power of a polynomial.{{sfnp|Heninger|Rains|Sloane|2006}}

As with other classes of [[smooth number]]s, regular numbers are important as problem sizes in computer programs for performing the [[fast Fourier transform]], a technique for analyzing the dominant frequencies of signals in [[Signal (electrical engineering)|time-varying data]]. For instance, the method of {{harvtxt|Temperton|1992}} requires that the transform length be a regular number.{{sfnp|Temperton|1992}}

Book VIII of [[Plato]]'s ''[[The Republic (Plato)|Republic]]'' involves an allegory of marriage centered on the highly regular number 60<sup>4</sup> = 12,960,000 and its divisors (see [[Plato's number]]). Later scholars have invoked both Babylonian mathematics and music theory in an attempt to explain this passage.<ref>{{harvtxt|Barton|1908}}; {{harvtxt|McClain|1974}}.</ref>

Certain species of [[bamboo]] release large numbers of seeds in synchrony (a process called [[Mast (botany)|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.{{sfnp|Veller|Nowak|Davis|2015}} 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.{{sfnp|Veller|Nowak|Davis|2015}}

== Notes ==
{{Reflist}}

== References ==
{{refbegin|30em}}
*{{citation
 | author-link = Aaboe
 | last = Aaboe | first = Asger
 | title = Some Seleucid mathematical tables (extended reciprocals and squares of regular numbers)
 | journal = [[Journal of Cuneiform Studies]]
 | volume = 19
 | issue = 3
 | pages = 79–86
 | year = 1965
 | mr = 0191779 
 | doi = 10.2307/1359089
 | publisher = The American Schools of Oriental Research
 | jstor = 1359089| s2cid = 164195082 }}.
*{{citation
 | last = Asmussen
 | first = Robert
 | title = Periodicity of sinusoidal frequencies as a basis for the analysis of Baroque and Classical harmony: a computer based study
 | year = 2001
 | url = http://www.terraworld.net/c-jasmussen/thesis_asmussen.pdf
 | series = Ph.D. thesis
 | publisher = [[University of Leeds]]
 | access-date = 2007-03-15
 | archive-date = 2016-04-24
 | archive-url = https://web.archive.org/web/20160424195738/http://www.terraworld.net/c-jasmussen/thesis_asmussen.pdf
 | url-status = dead
 }}.
*{{citation
 | last = Barton | first = George A.
 | title = On the Babylonian origin of Plato's nuptial number
 | year = 1908
 | journal = [[Journal of the American Oriental Society]]
 | volume = 29
 | pages = 210–219
 | doi = 10.2307/592627
 | publisher = American Oriental Society
 | jstor = 592627}}.
*{{citation|title=Ramanujan: letters and commentary|volume=9|series=History of mathematics|publisher=American Mathematical Society|editor1-first=Bruce C.|editor1-last=Berndt|editor2-first=Robert Alexander|editor2-last=Rankin|year=1995|isbn=978-0-8218-0470-4|page=23|bibcode=1995rlc..book.....B}}.
*{{citation
 | last = Bruins | first = E. M.
 | contribution = La construction de la grande table le valeurs réciproques AO 6456
 | title = Actes de la XVII<sup>e</sup> Rencontre Assyriologique Internationale
 | editor-last = Finet | editor-first = André
 | publisher = Comité belge de recherches en Mésopotamie
 | year = 1970
 | pages = 99–115}}.
*{{citation
 | last = Clarke | first = A. R.
 | date = January 1877
 | doi = 10.1038/015253b0
 | issue = 377
 | journal = Nature
 | pages = 253
 | title = Just intonation
 | volume = 15| bibcode = 1877Natur..15..253C
 | doi-access = free
 }}.
*{{citation
 | author-link1 = John H. Conway
 | last1 = Conway
 | first1 = John H.
 | author-link2 = Richard K. Guy
 | last2 = Guy
 | first2 = Richard K.
 | title = The Book of Numbers
 | pages = [https://archive.org/details/bookofnumbers0000conw/page/172 172–176]
 | publisher = Copernicus
 | year = 1996
 | isbn = 0-387-97993-X
 | url = https://archive.org/details/bookofnumbers0000conw/page/172
 }}.
*{{citation
 | author-link = Edsger W. Dijkstra
 | last = Dijkstra
 | first = Edsger W.
 | contribution = 17. An exercise attributed to R. W. Hamming
 | isbn = 978-0132158718
 | pages = [https://archive.org/details/disciplineofprog0000dijk/page/129 129–134]
 | publisher = Prentice-Hall
 | title = A Discipline of Programming
 | url = https://archive.org/details/disciplineofprog0000dijk/page/129
 | year = 1976
 }}
*{{citation
 | author-link = Edsger W. Dijkstra
 | last = Dijkstra
 | first = Edsger W.
 | title = Hamming's exercise in SASL
 | year = 1981
 | url = http://www.cs.utexas.edu/users/EWD/ewd07xx/EWD792.PDF
 | id = Report EWD792. Originally a privately circulated handwritten note
 }}.
*{{citation 
 |author-link=David Eppstein 
 |last=Eppstein 
 |first=David 
 |title=The range-restricted Hamming problem 
 |year=2007 
 |url=https://11011110.github.io/blog/2007/03/18/range-restricted-hamming-problem.html 
 }}.
*{{citation
 | last1=Fowler
 | first1=David
 | last2=Robson
 | first2=Eleanor
 | title=Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context
 | journal=Historia Mathematica
 | volume=25
 | date=1998
 | issue=4
 | pages=366–378
 | doi=10.1006/hmat.1998.2209
 | url=https://www.sciencedirect.com/science/article/pii/S0315086098922091/pdf?md5=7c3e116fac2deef21b107b78f3561862&pid=1-s2.0-S0315086098922091-main.pdf
 | doi-access=free
 }}, page 375.
*{{citation
 | last = Gingerich | first = Owen
 | title = Eleven-digit regular sexagesimals and their reciprocals
 | year = 1965
 | journal = [[Transactions of the American Philosophical Society]]
 | volume = 55
 | issue = 8
 | pages = 3–38
 | doi = 10.2307/1006080
 | publisher = American Philosophical Society
 | jstor = 1006080}}.
*{{citation
 | last = Habens
 | first = Rev. W. J.
 | title = On the musical scale
 | journal = Proceedings of the Musical Association
 | year = 1889
 | pages = 16th Session, p. 1
 | jstor = 765355
 | volume = 16
 | publisher = Royal Musical Association
 | url = https://zenodo.org/record/2404277
 }}.
*{{citation
 | last1 = Halsey | first1 = G. D. | last2 = Hewitt | first2 = Edwin
 | title = More on the superparticular ratios in music
 | journal = [[American Mathematical Monthly]]
 | volume = 79
 | year = 1972
 | pages = 1096–1100
 | mr = 0313189 
 | doi = 10.2307/2317424
 | issue = 10
 | publisher = Mathematical Association of America
 | jstor = 2317424}}.
*{{citation
 | last = Hemmendinger | first = David
 | journal = [[ACM SIGPLAN Notices]]
 | year = 1988
 | title = The "Hamming problem" in Prolog
 | volume = 23
 | issue = 4
 | pages = 81–86
 | doi = 10.1145/44326.44335| s2cid = 28906392
 }}.
*{{citation
 | last1 = Heninger | first1 = Nadia | author1-link = Nadia Heninger
 | last2 = Rains | first2 = E. M.
 | last3 = Sloane | first3 = N. J. A. | author3-link = Neil Sloane
 | arxiv = math.NT/0509316
 | doi = 10.1016/j.jcta.2006.03.018
 | issue = 8
 | journal = [[Journal of Combinatorial Theory]]
 | mr = 2269551
 | pages = 1732–1745
 | series = Series A
 | title = On the integrality of {{mvar|n}}th roots of generating functions
 | volume = 113
 | year = 2006| s2cid = 15913795 }}}.
*{{citation
 | last1 = Honingh | first1 = Aline | last2 = Bod | first2 = Rens
 | title = Convexity and the well-formedness of musical objects
 | journal = [[Journal of New Music Research]]
 | year = 2005
 | volume = 34
 | issue = 3
 | pages = 293–303
 | doi = 10.1080/09298210500280612| s2cid = 16321292 }}.
*{{citation
 | last1 = Howard | first1 = Deborah | author1-link = Deborah Howard
 | last2 = Longair | first2 = Malcolm
 | date = May 1982
 | doi = 10.2307/989675
 | issue = 2
 | journal = Journal of the Society of Architectural Historians
 | jstor = 989675
 | pages = 116–143
 | title = Harmonic proportion and Palladio's ''Quattro Libri''
 | volume = 41}}
*{{citation | last = Knuth | first = D. E. | author-link = Donald Knuth | title = Ancient Babylonian algorithms | journal = [[Communications of the ACM]] | volume = 15 | issue = 7 | year = 1972 | pages = 671–677 | doi = 10.1145/361454.361514 | s2cid = 7829945 | url = https://teaching.csse.uwa.edu.au/units/CITS1001/extension/ancient-babylonian-algorithms.pdf }}. A correction appears in [https://dl.acm.org/doi/pdf/10.1145/359997.360014 ''CACM'' '''19'''(2), 1976], stating that the tablet does not contain all 231 of the numbers of interest. The article (corrected) with a brief addendum appears in [https://doc.lagout.org/science/0_Computer%20Science/7_Technical%20Papers/Selected%20Papers%20on%20Computer%20Science%20-%20Donald%20E.%20Knuth.pdf ''Selected Papers on Computer Science'', CSLI Lecture Notes 59, Cambridge Univ. Press, 1996, pp.&nbsp;185–203], but without the Appendix that was included in the original paper.
*{{citation
 | last = Kopiez | first = Reinhard
 | doi = 10.1525/mp.2003.20.4.383
 | issue = 4
 | journal = Music Perception
 | pages = 383–410
 | title = Intonation of harmonic intervals: adaptability of expert musicians to equal temperament and just intonation
 | volume = 20
 | year = 2003}}
*{{citation
 | last = Longuet-Higgins | first = H. C. | author-link = H. Christopher Longuet-Higgins
 | year = 1962
 | title = Letter to a musical friend
 | journal = Music Review
 | issue = August
 | pages = 244–248}}.
*{{citation
 | last = McClain | first = Ernest G.
 | author-link = Ernest G. McClain
 | year = 1974
 | title = Musical "Marriages" in Plato's "Republic"
 | journal = [[Journal of Music Theory]]
 | volume = 18
 | issue = 2
 | pages = 242–272
 | jstor = 843638
 | publisher = Duke University Press}}.
*{{citation
 | last = Pomerance | first = Carl | author-link = Carl Pomerance
 | contribution = The role of smooth numbers in number-theoretic algorithms
 | mr = 1403941
 | pages = 411–422
 | publisher = Birkhäuser | location = Basel
 | title = Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994)
 | year = 1995}}.
*{{citation
 | last = Sachs | first = A. J.
 | year = 1947
 | title = Babylonian mathematical texts. I. Reciprocals of regular sexagesimal numbers
 | journal = [[Journal of Cuneiform Studies]]
 | volume = 1
 | issue = 3
 | pages = 219–240
 | mr = 0022180 
 | doi = 10.2307/1359434
 | publisher = The American Schools of Oriental Research
 | jstor = 1359434| s2cid = 163783242
 }}.
*{{citation
 | last = Silver | first = A. L. Leigh
 | title = Musimatics or the nun's fiddle
 | journal = [[American Mathematical Monthly]]
 | volume = 78
 | year = 1971
 | issue = 4
 | pages = 351–357
 | doi = 10.2307/2316896
 | publisher = Mathematical Association of America
 | jstor = 2316896}}.
*{{Cite OEIS|A051037|5-smooth numbers|mode=cs2|ref={{harvid|Sloane "A051037"}}}}
*{{citation
 | last = Størmer | first = Carl
 | author-link = Carl Størmer
 | title = Quelques théorèmes sur l'équation de Pell {{math|1=''x''<sup>2</sup> &minus; ''Dy''<sup>2</sup> = ±1}} et leurs applications
 | journal = Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl.
 | volume = I
 | issue = 2
 | year = 1897}}.
*{{citation
 | last = Temperton | first = Clive
 | title = A generalized prime factor FFT algorithm for any {{math|1=''N'' = 2<sup>''p''</sup>3<sup>''q''</sup>5<sup>''r''</sup>}}
 | journal = SIAM Journal on Scientific and Statistical Computing
 | volume = 13
 | issue = 3
 | year = 1992
 | pages = 676–686
 | doi = 10.1137/0913039| s2cid = 14764894
 }}.
*{{citation
 | last1 = Veller | first1 = Carl
 | last2 = Nowak | first2 = Martin A.
 | last3 = Davis | first3 = Charles C.
 | date = May 2015
 | doi = 10.1111/ele.12442
 | issue = 7
 | journal = Ecology Letters
 | pages = 653–659
 | title = Extended flowering intervals of bamboos evolved by discrete multiplication
 | volume = 18| pmid = 25963600
 | bibcode = 2015EcolL..18..653V
 }}
*{{citation
 | last = Wolf | first = Daniel James
 | date = March 2003
 | doi = 10.1080/0749446032000134715
 | issue = 1–2
 | journal = Contemporary Music Review
 | pages = 3–14
 | title = Alternative tunings, alternative tonalities
 | volume = 22| s2cid = 191457676
 }}
*{{citation
 | last = Yuen | first = C. K.
 | title = Hamming numbers, lazy evaluation, and eager disposal
 | journal = [[ACM SIGPLAN Notices]]
 | year = 1992
 | volume = 27
 | issue = 8
 | pages = 71–75
 | doi = 10.1145/142137.142151| s2cid = 18283005
 }}.
{{refend}}

==External links==
*[http://aleph0.clarku.edu/~djoyce/ma105/reciprocals3600.html Table of reciprocals of regular numbers up to 3600] from the web site of Professor David E. Joyce, Clark University.
*[http://rosettacode.org/wiki/Hamming_numbers RosettaCode] Generation of Hamming_numbers in ~ 50 programming languages

{{Divisor classes}}
{{Classes of natural numbers}}

[[Category:Babylonian mathematics]]
[[Category:Integer sequences]]
[[Category:Functional programming]]
[[Category:Mathematics of music]]