正规数 (整数)

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

一个400以内正规数其因数关系的哈斯图，其纵向为对数尺度^[1]

在许多数学及应用的领域会用到60乘幂的因数，在不同的领域中其名称也有所不同。

• 在数论中，60乘幂的因数也称为5-光滑数，因为其质因数只有2,3或是5，这是k-光滑数中的一个特例，k-光滑数是指其质因数都小于等于k的整数。
• 在巴比伦数学中，60乘幂的因数称为正规数或是60正规数，因为巴比伦数学是使用六十进制，因此这类数字格外的重要。
• 在计算机科学，60乘幂的因数称为汉明数（Hamming numbers），得名自数学家理查德·卫斯里·汉明，他提出一个用电脑依序找出60乘幂的因数的演算法。

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, Periodicity of sinusoidal frequencies as a basis for the analysis of Baroque and Classical harmony: a computer based study (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, （原始内容存档 (PDF)于2019-04-04）.
• Eppstein, David, The range-restricted Hamming problem, 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.