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。 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。 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）. 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 XVIIe 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 = 2p3q5r, 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 .