正规数 (整数)

正规数（Regular numbers）是指可以整除60的乘幂的整數，也就是60乘幂的的因數，例如60² = 3600 = 48 × 75，48和75都可以整除60的平方，也都是正规数。

一個400以內正规数其因數關係的哈斯圖，其縱向為對數尺度^[1]

在許多數學及應用的領域會用到60乘幂的因數，在不同的領域中其名稱也有所不同。

• 在數論中，60乘幂的因數也稱為5-光滑數，因為其質因數只有2,3或是5，這是k-光滑數中的一個特例，k-光滑數是指其質因數都小於等於k的整數。
• 在巴比伦数学中，60乘幂的因數稱為正规数或是60正规数，因為巴比伦数学是使用六十進制，因此這類數字格外的重要。
• 在計算機科學，60乘幂的因數稱為漢明數（Hamming numbers），得名自數學家理查德·衛斯里·漢明，他提出一個用電腦依序找出60乘幂的因數的演算法。

注释

1. Inspired by similar diagrams by Erkki Kurenniemi in "Chords, scales, and divisor lattices" （页面存档备份，存于互联网档案馆）.

参考资料

• Aaboe, Asger, Some Seleucid mathematical tables (extended reciprocals and squares of regular numbers), Journal of Cuneiform Studies (The American Schools of Oriental Research), 1965, 19 (3): 79–86, JSTOR 1359089, MR 0191779, doi:10.2307/1359089 .
• Asmussen, Robert, (PDF), Ph.D. thesis, Univ. of Leeds, 2001 [2012-12-27], （原始内容 (PDF)存档于2016-04-24） .
• Barton, George A., On the Babylonian origin of Plato's nuptial number, Journal of the American Oriental Society (American Oriental Society), 1908, 29: 210–219, JSTOR 592627, doi:10.2307/592627 .
• Bruins, E. M., La construction de la grande table le valeurs réciproques AO 6456, Finet, André (编), Actes de la XVII^e Rencontre Assyriologique Internationale, Comité belge de recherches en Mésopotamie: 99–115, 1970 .
• Conway, John H.; Guy, Richard K., The Book of Numbers, Copernicus: 172–176, 1996, ISBN 0-387-97993-X .
• Dijkstra, Edsger W., Hamming's exercise in SASL (PDF), 1981 [2012-12-27], Report EWD792. Originally a privately-circulated handwitten note, （原始内容 (PDF)于2019-04-04） .
• Eppstein, David, , 2007, （原始内容存档于2011-07-21） .
• Gingerich, Owen, Eleven-digit regular sexagesimals and their reciprocals, Transactions of the American Philosophical Society (American Philosophical Society), 1965, 55 (8): 3–38, JSTOR 1006080, doi:10.2307/1006080 .
• Habens, Rev. W. J., On the musical scale, Proceedings of the Musical Association (Royal Musical Association), 1889, 16: 16th Session, p. 1, JSTOR 765355 .
• Halsey, G. D.; Hewitt, Edwin, More on the superparticular ratios in music, American Mathematical Monthly (Mathematical Association of America), 1972, 79 (10): 1096–1100, JSTOR 2317424, MR 0313189, doi:10.2307/2317424 .
• Hemmendinger, David, The "Hamming problem" in Prolog, ACM SIGPLAN Notices, 1988, 23 (4): 81–86, doi:10.1145/44326.44335 .
• Heninger, Nadia; Rains, E. M.; Sloane, N. J. A. On the integrality of nth roots of generating functions. 2005. arXiv:math.NT/0509316  . .
• Honingh, Aline; Bod, Rens, Convexity and the well-formedness of musical objects, Journal of New Music Research, 2005, 34 (3): 293–303, doi:10.1080/09298210500280612 .
• Knuth, D. E., Ancient Babylonian algorithms, Communications of the ACM, 1972, 15 (7): 671–677, doi:10.1145/361454.361514 . Errata in CACM 19(2), 1976. Reprinted with a brief addendum in Selected Papers on Computer Science, CSLI Lecture Notes 59, Cambridge Univ. Press, 1996, pp. 185–203.
• Longuet-Higgins, H. C., Letter to a musical friend, Music Review, 1962, (August): 244–248 .
• McClain, Ernest G.; Plato, Musical "Marriages" in Plato's "Republic", Journal of Music Theory (Duke University Press), 1974, 18 (2): 242–272, JSTOR 843638, doi:10.2307/843638 .
• Sachs, A. J., Babylonian mathematical texts. I. Reciprocals of regular sexagesimal numbers, Journal of Cuneiform Studies (The American Schools of Oriental Research), 1947, 1 (3): 219–240, JSTOR 1359434, MR 0022180, doi:10.2307/1359434 .
• Silver, A. L. Leigh, Musimatics or the nun's fiddle, American Mathematical Monthly (Mathematical Association of America), 1971, 78 (4): 351–357, JSTOR 2316896, doi:10.2307/2316896 .
• Størmer, Carl, Quelques théorèmes sur l'équation de Pell x² - Dy² = ±1 et leurs applications, Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl., 1897, I (2) .
• Temperton, Clive, A generalized prime factor FFT algorithm for any N = 2^p3^q5^r, SIAM Journal on Scientific and Statistical Computing, 1992, 13 (3): 676–686, doi:10.1137/0913039 .
• Yuen, C. K., Hamming numbers, lazy evaluation, and eager disposal, ACM SIGPLAN Notices, 1992, 27 (8): 71–75, doi:10.1145/142137.142151 .