989:
2052:) 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. 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.
3485:
40:
5746:
1170:
Thus, 1/54, in sexagesimal, is 1/60 + 6/60 + 40/60, also denoted 1:6:40 as
Babylonian notational conventions did not specify the power of the starting digit. Conversely 1/4000 = 54/60, so division by 1:6:40 = 4000 can be accomplished by instead multiplying by 54 and shifting three sexagesimal places.
1177:
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 (2 1 4 8 3 0 7 in sexagesimal), whose reciprocal has 17 sexagesimal
1169:
For instance, consider division by the regular number 54 = 23. 54 is a divisor of 60, and 60/54 = 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.
1182:
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
1186:
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 and the broken tablet
909:
1356:
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. Thus, for an instrument with this tuning, all pitches are regular-number
1772:, 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
769:
1380:
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.
367:
1282:
666:
607:
471:
1389:
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
244:
2713:
3042:
1376:
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:
1173:
The
Babylonians used tables of reciprocals of regular numbers, some of which still survive. These tables existed relatively unchanged throughout Babylonian times. One tablet from
3848:
1480:
798:
1404:
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
950:
1393:. 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
2744:
2244:, 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
1876:
1116:
1440:
529:
1882:
for an early description of computer code that generates these numbers out of order and then sorts them; Knuth describes an ad hoc algorithm, which he attributes to
1843:
1053:
1164:
1081:
1759:
1736:
1713:
1643:
1620:
1597:
2003:
1979:
1959:
1928:
1904:
1812:
1686:
1663:
1322:
1302:
1136:
1026:
970:
793:
686:
491:
304:
284:
264:
3428:
3219:
695:
408:
401:
3841:
2524:
1818:
above). In algorithmic terms, this is equivalent to generating (in order) the subsequence of the infinite sequence of regular numbers, ranging from
1324:
both regular and less than 60. Fowler and Robson discuss the calculation of square roots, such as how the
Babylonians found an approximation to the
988:
1373:
between any two pitches can be described as a product 235 of powers of the prime numbers up to 5, or equivalently as a ratio of regular numbers.
85:
These numbers arise in several areas of mathematics and its applications, and have different names coming from their different areas of study.
2570:
1787:, lazy functional code for generating regular numbers is used as one of the built-in tests for correctness of the language's implementation.
3292:
Veller, Carl; Nowak, Martin A.; Davis, Charles C. (May 2015), "Extended flowering intervals of bamboos evolved by discrete multiplication",
414:
Although the regular numbers appear dense within the range from 1 to 60, they are quite sparse among the larger integers. A regular number
4648:
3834:
3052:
Kopiez, Reinhard (2003), "Intonation of harmonic intervals: adaptability of expert musicians to equal temperament and just intonation",
3041:, stating that the tablet does not contain all 231 of the numbers of interest. The article (corrected) with a brief addendum appears in
2491:
4643:
2671:
1561:) attributes to Hamming the problem of building the infinite ascending sequence of all 5-smooth numbers; this problem is now known as
4658:
4638:
1401:
of modern pianos is not a 5-limit tuning, and some modern composers have experimented with tunings based on primes larger than five.
538:
82:. As an example, 60 = 3600 = 48 × 75, so as divisors of a power of 60 both 48 and 75 are regular.
3474:
2643:
1533:
of buildings. In connection with the analysis of these shared musical and architectural ratios, for instance in the architecture of
5351:
4931:
3671:
3484:
1777:
5780:
5770:
2881:
2017:
1773:
1769:
2998:
1886:, for generating the six-digit numbers more quickly but that does not generalize in a straightforward way to larger values of
4653:
3072:
5437:
135:(base 60) number system that the Babylonians used for writing their numbers, and that was central to Babylonian mathematics.
5785:
3421:
3180:
2930:
2800:
2493:
Periodicity of sinusoidal frequencies as a basis for the analysis of
Baroque and Classical harmony: a computer based study
1784:
4753:
5103:
4422:
4215:
3133:
2445:
309:
5138:
5108:
4783:
4773:
3625:
1201:
612:
5279:
4693:
4427:
4407:
4969:
5133:
5775:
5228:
4851:
4608:
4417:
4399:
4293:
4283:
4273:
3661:
2702:
2041:). Later scholars have invoked both Babylonian mathematics and music theory in an attempt to explain this passage.
417:
5113:
2580:
Bruins, E. M. (1970), "La construction de la grande table le valeurs réciproques AO 6456", in Finet, André (ed.),
5356:
4901:
4522:
4308:
4303:
4298:
4288:
4265:
3646:
3006:
1394:
196:
4341:
3414:
4598:
394:
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, ... (sequence
2037:
involves an allegory of marriage centered on the highly regular number 60 = 12,960,000 and its divisors (see
5467:
5432:
5218:
5128:
5002:
4977:
4886:
4876:
4488:
4470:
4390:
3800:
3666:
3590:
3090:
2293:
1515:
5727:
4997:
4871:
4502:
4278:
4058:
3985:
3651:
3610:
2343:
2013:
1511:
1443:
1005:
177:
120:
4982:
4836:
4763:
3918:
3580:
3449:
1362:
5691:
5331:
1448:
5624:
5518:
5482:
5223:
4946:
4926:
4743:
4412:
4200:
4172:
3754:
3656:
3301:
3131:
Sachs, A. J. (1947), "Babylonian mathematical texts. I. Reciprocals of regular sexagesimal numbers",
2599:
2558:
2033:
914:
5346:
5210:
5205:
5173:
4936:
4911:
4906:
4881:
4811:
4807:
4738:
4628:
4460:
4256:
4225:
3815:
3810:
3605:
3600:
3585:
3524:
3358:
2839:
2500:
2443:(1965), "Some Seleucid mathematical tables (extended reciprocals and squares of regular numbers)",
973:
5745:
5749:
5503:
5498:
5412:
5386:
5284:
5263:
5035:
4916:
4866:
4788:
4758:
4698:
4465:
4445:
4376:
4089:
3739:
3734:
3695:
3615:
3595:
3375:
3345:
3280:
3197:
3166:
3150:
3099:
3023:
2983:
2947:
2916:
2890:
2856:
2817:
2786:
2761:
2680:
2653:
2541:
2478:
2462:
1982:
1526:
1196:
1192:
1118:, shifted by some number of places. This allows for easy division by these numbers: to divide by
4633:
2663:
2657:
1910:
describes an algorithm for computing tables of this type in linear time for arbitrary values of
176:
for generating these numbers in ascending order. This problem has been used as a test case for
39:
5643:
5588:
5442:
5417:
5391:
5168:
4846:
4841:
4768:
4748:
4733:
4455:
4437:
4356:
4346:
4331:
4109:
4094:
3775:
3715:
3394:
3317:
3085:
2667:
2639:
2566:
2038:
1848:
1398:
1086:
143:
52:
2635:
2629:
1407:
496:
5679:
5472:
5058:
5030:
5020:
5012:
4896:
4861:
4856:
4823:
4517:
4480:
4371:
4366:
4361:
4351:
4323:
4210:
4162:
4157:
4114:
4053:
3805:
3780:
3700:
3686:
3620:
3504:
3464:
3367:
3337:
3309:
3272:
3189:
3142:
3061:
3015:
2975:
2939:
2900:
2848:
2809:
2753:
2728:
2607:
2533:
2504:
2454:
1821:
1499:
1495:
1370:
1325:
1031:
161:
3162:
3123:
2912:
2829:
2474:
5655:
5544:
5477:
5403:
5326:
5300:
5118:
4831:
4688:
4623:
4593:
4583:
4578:
4244:
4152:
4099:
3943:
3883:
3790:
3785:
3710:
3704:
3641:
3539:
3529:
3459:
3158:
3119:
2928:
Honingh, Aline; Bod, Rens (2005), "Convexity and the well-formedness of musical objects",
2908:
2825:
2470:
1766:
1689:
1550:
1507:
1503:
1341:
904:{\displaystyle {\frac {\left(\ln(N{\sqrt {30}})\right)^{3}}{6\ln 2\ln 3\ln 5}}+O(\log N),}
169:
146:
3228:
2012:, regular numbers are important as problem sizes in computer programs for performing the
1141:
1058:
3305:
2684:
2603:
2562:
1741:
1718:
1695:
1625:
1602:
1579:
5660:
5528:
5513:
5377:
5341:
5316:
5192:
5163:
5025:
4921:
4891:
4618:
4573:
4450:
4048:
4043:
4038:
4010:
3995:
3908:
3893:
3871:
3858:
3795:
3749:
3575:
3559:
3519:
3111:
2959:
2868:
2698:
2625:
2621:
1988:
1964:
1944:
1913:
1889:
1797:
1671:
1648:
1519:
1491:
1345:
1307:
1287:
1121:
1011:
955:
778:
772:
671:
476:
289:
269:
249:
3406:
2203:
has other interpretations, for which see its article, but all involve regular numbers.
1549:
Algorithms for calculating the regular numbers in ascending order were popularized by
5764:
5583:
5567:
5508:
5462:
5158:
5143:
5053:
4778:
4336:
4205:
4167:
4124:
4005:
3990:
3980:
3938:
3928:
3903:
3744:
3544:
3534:
3514:
3349:
3170:
3116:
Proceedings of the
International Congress of Mathematicians, Vol. 1, 2 (ZĂĽrich, 1994)
2482:
2049:
2009:
1487:
1366:
1349:
370:
106:
90:
44:
33:
3379:
3284:
2951:
2920:
2860:
2798:
Halsey, G. D.; Hewitt, Edwin (1972), "More on the superparticular ratios in music",
5619:
5608:
5523:
5361:
5336:
5253:
5153:
5123:
5098:
5082:
4987:
4954:
4703:
4677:
4588:
4527:
4104:
4000:
3933:
3913:
3888:
3759:
3676:
3554:
3499:
3469:
3032:
3027:
2994:
2742:
Gingerich, Owen (1965), "Eleven-digit regular sexagesimals and their reciprocals",
2200:
1530:
1390:
1337:
1188:
1179:
374:
150:
139:
98:
48:
2557:, History of mathematics, vol. 9, American Mathematical Society, p. 23,
1178:
digits. Noting the difficulty of both calculating these numbers and sorting them,
911:
and it has been conjectured that the error term of this approximation is actually
5578:
5453:
5258:
4722:
4613:
4568:
4563:
4313:
4220:
4119:
3948:
3923:
3898:
3341:
3328:
Wolf, Daniel James (March 2003), "Alternative tunings, alternative tonalities",
3209:
2872:
2522:
Barton, George A. (1908), "On the
Babylonian origin of Plato's nuptial number",
1001:
977:
532:
132:
67:
63:
2904:
2714:"Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context"
2069:
17:
5715:
5696:
4992:
4603:
2943:
2248:. In the modern music theory literature this assertion is often attributed to
3826:
3065:
5321:
5248:
5240:
5045:
4959:
4077:
3509:
3356:
Yuen, C. K. (1992), "Hamming numbers, lazy evaluation, and eager disposal",
173:
3321:
3256:
Temperton, Clive (1992), "A generalized prime factor FFT algorithm for any
2733:
1776:
sequential implementations are also possible whereas explicitly concurrent
3371:
3019:
1328:, perhaps using regular number approximations of fractions such as 17/12.
5422:
3400:
1961:
is a regular number and is divisible by 8, the generating function of an
1534:
1358:
1174:
51:
relationships among the regular numbers up to 400. The vertical scale is
2895:
2852:
2110:
5427:
5086:
3454:
3201:
3154:
2821:
2765:
2466:
2253:
1482:
that is meaningful as a musical interval. These intervals are 2/1 (the
1386:
190:
97:, because they can be characterized as having only 2, 3, or 5 as their
3313:
3103:
3046:, CSLI Lecture Notes 59, Cambridge Univ. Press, 1996, pp. 185–203
2987:
2790:
2545:
3114:(1995), "The role of smooth numbers in number-theoretic algorithms",
2612:
2045:
1483:
1353:
689:
70:). Equivalently, they are the numbers whose only prime divisors are
3276:
3193:
3146:
2813:
2774:
2757:
2458:
2082:
2080:
2078:
609:
as can be seen by taking logarithms of both sides of the inequality
3048:, but without the Appendix that was included in the original paper.
2979:
2962:; Longair, Malcolm (May 1982), "Harmonic proportion and Palladio's
2692:, Report EWD792. Originally a privately circulated handwritten note
2537:
2016:, a technique for analyzing the dominant frequencies of signals in
3725:
2440:
2028:
1382:
987:
38:
3397:
from the web site of
Professor David E. Joyce, Clark University.
1529:, musical ratios were used in other applications, including the
993:
764:{\displaystyle {\frac {\log _{2}N\,\log _{3}N\,\log _{5}N}{6}}.}
5713:
5677:
5641:
5605:
5565:
5190:
5079:
4805:
4720:
4675:
4552:
4242:
4189:
4141:
4075:
4027:
3965:
3869:
3830:
3410:
2837:
Hemmendinger, David (1988), "The "Hamming problem" in Prolog",
2418:
2416:
1569:. Dijkstra's ideas to compute these numbers are the following:
3178:
Silver, A. L. Leigh (1971), "Musimatics or the nun's fiddle",
2048:
release large numbers of seeds in synchrony (a process called
3244:
Skrifter
Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl.
952:. A similar formula for the number of 3-smooth numbers up to
668:. Therefore, the number of regular numbers that are at most
381:-smooth number is a number whose greatest prime factor is at
2584:, Comité belge de recherches en Mésopotamie, pp. 99–115
3403:
Generation of Hamming_numbers in ~ 50 programming languages
3213:
1765:
This algorithm is often used to demonstrate the power of a
396:
2252:, who used a graphical arrangement closely related to the
2179:
2177:
149:. In connection with music theory and related theories of
131:, and are of great importance in this area because of the
3141:(3), The American Schools of Oriental Research: 219–240,
2553:
Berndt, Bruce C.; Rankin, Robert Alexander, eds. (1995),
2225:
2223:
2221:
1573:
The sequence of Hamming numbers begins with the number 1.
79:
75:
71:
2582:
Actes de la XVII Rencontre Assyriologique Internationale
2024:
requires that the transform length be a regular number.
992:
AO 6456, a table of reciprocals of regular numbers from
2656:(1976), "17. An exercise attributed to R. W. Hamming",
2453:(3), The American Schools of Oriental Research: 79–86,
2808:(10), Mathematical Association of America: 1096–1100,
2086:
1453:
3088:(1974), "Musical "Marriages" in Plato's "Republic"",
1991:
1967:
1947:
1916:
1892:
1851:
1824:
1800:
1744:
1721:
1698:
1688:
may be generated by outputting the value 1, and then
1674:
1651:
1628:
1605:
1582:
1576:
The remaining values in the sequence are of the form
1451:
1410:
1310:
1290:
1204:
1144:
1124:
1089:
1061:
1034:
1014:
958:
917:
801:
781:
698:
674:
615:
541:
499:
479:
420:
312:
292:
272:
252:
199:
3265:
SIAM Journal on Scientific and Statistical Computing
2068:
Inspired by similar diagrams by Erkki Kurenniemi in
1008:
of a regular number has a finite representation. If
5537:
5491:
5451:
5402:
5376:
5309:
5293:
5272:
5239:
5204:
5044:
5011:
4968:
4945:
4822:
4510:
4501:
4479:
4436:
4398:
4389:
4322:
4264:
4255:
3768:
3724:
3685:
3634:
3568:
3492:
3442:
3231:(1897), "Quelques théorèmes sur l'équation de Pell
3188:(4), Mathematical Association of America: 351–357,
2379:
2164:
2162:
2160:
2135:
2133:
2131:
1938:
1814:-digit sexagesimal numbers in ascending order (see
1565:, and the numbers so generated are also called the
3395:Table of reciprocals of regular numbers up to 3600
2968:Journal of the Society of Architectural Historians
2745:Transactions of the American Philosophical Society
1997:
1973:
1953:
1922:
1898:
1870:
1837:
1806:
1753:
1730:
1707:
1680:
1657:
1637:
1614:
1591:
1474:
1434:
1316:
1296:
1276:
1158:
1130:
1110:
1075:
1047:
1020:
964:
944:
903:
787:
763:
680:
660:
601:
523:
485:
465:
362:{\displaystyle 60^{\max(\lceil i\,/2\rceil ,j,k)}}
361:
298:
278:
258:
238:
142:, regular numbers occur in the ratios of tones in
2785:, Royal Musical Association: 16th Session, p. 1,
2590:Clarke, A. R. (January 1877), "Just intonation",
1277:{\displaystyle (p^{2}-q^{2},\,2pq,\,p^{2}+q^{2})}
661:{\displaystyle 2^{i}\cdot 3^{j}\cdot 5^{k}\leq N}
2422:
2199:for a popular treatment of this interpretation.
2111:OEIS search for sequences involving 5-smoothness
1537:, the regular numbers have also been called the
318:
109:, the numbers that have no prime factor greater
3214:"Sequence A051037 (5-smooth numbers)"
2300:), and provide a proof for this case; see also
2249:
602:{\displaystyle i\ln 2+j\ln 3+k\ln 5\leq \ln N,}
535:bounded by the coordinate planes and the plane
172:, who proposed the problem of finding computer
101:. This is a specific case of the more general
3842:
3422:
2313:
466:{\displaystyle n=2^{i}\cdot 3^{j}\cdot 5^{k}}
411:have definitions involving 5-smooth numbers.
8:
2326:
2289:
2183:
2122:
339:
324:
2752:(8), American Philosophical Society: 3–38,
1815:
239:{\displaystyle 2^{i}\cdot 3^{j}\cdot 5^{k}}
5710:
5674:
5638:
5602:
5562:
5236:
5201:
5187:
5076:
4819:
4802:
4717:
4672:
4549:
4507:
4395:
4261:
4252:
4239:
4186:
4143:Possessing a specific set of other numbers
4138:
4072:
4024:
3962:
3866:
3849:
3835:
3827:
3429:
3415:
3407:
2229:
1377:
123:, the divisors of powers of 60 are called
3220:On-Line Encyclopedia of Integer Sequences
2894:
2732:
2611:
2391:
2355:
2196:
2098:
2021:
1990:
1966:
1946:
1915:
1891:
1879:
1856:
1850:
1829:
1823:
1799:
1743:
1720:
1697:
1673:
1650:
1627:
1604:
1581:
1452:
1450:
1409:
1309:
1289:
1265:
1252:
1247:
1234:
1225:
1212:
1203:
1148:
1143:
1123:
1100:
1094:
1088:
1065:
1060:
1055:, then the sexagesimal representation of
1039:
1033:
1013:
957:
916:
837:
822:
802:
800:
780:
740:
735:
723:
718:
706:
699:
697:
673:
646:
633:
620:
614:
540:
498:
478:
457:
444:
431:
419:
409:On-Line Encyclopedia of Integer Sequences
331:
330:
317:
311:
291:
271:
251:
230:
217:
204:
198:
62:are numbers that evenly divide powers of
2525:Journal of the American Oriental Society
2367:
2241:
1907:
1558:
1554:
473:is less than or equal to some threshold
369:. The regular numbers are also called 5-
3118:, Basel: Birkhäuser, pp. 411–422,
2712:Fowler, David; Robson, Eleanor (1998),
2407:
2297:
2061:
996:, copied from an unknown earlier source
29:Numbers that evenly divide powers of 60
3075:(1962), "Letter to a musical friend",
2779:Proceedings of the Musical Association
2532:, American Oriental Society: 210–219,
2403:
2301:
2265:
2245:
2212:
2070:"Chords, scales, and divisor lattices"
1883:
775:, the number of regular numbers up to
3098:(2), Duke University Press: 242–272,
2168:
2151:
2139:
1791:
153:, these numbers have been called the
7:
2704:The range-restricted Hamming problem
2330:
2277:
3437:Divisibility-based sets of integers
3044:Selected Papers on Computer Science
2879:th roots of generating functions",
2380:Heninger, Rains & Sloane (2006)
1939:Heninger, Rains & Sloane (2006)
164:, regular numbers are often called
391:The first few regular numbers are
25:
3475:Fundamental theorem of arithmetic
2555:Ramanujan: letters and commentary
1475:{\displaystyle {\tfrac {x+1}{x}}}
373:, indicating that their greatest
189:Formally, a regular number is an
5744:
5352:Perfect digit-to-digit invariant
3483:
2423:Veller, Nowak & Davis (2015)
1790:A related problem, discussed by
1780:solutions might be non-trivial.
377:is at most 5. More generally, a
306:. Such a number is a divisor of
2999:"Ancient Babylonian algorithms"
2882:Journal of Combinatorial Theory
2875:(2006), "On the integrality of
1770:functional programming language
407:Several other sequences at the
2020:. For instance, the method of
1429:
1411:
1348:involves regular numbers: the
1271:
1205:
945:{\displaystyle O(\log \log N)}
939:
921:
895:
883:
829:
816:
692:of this tetrahedron, which is
518:
500:
354:
321:
1:
4191:Expressible via specific sums
3181:American Mathematical Monthly
2931:Journal of New Music Research
2801:American Mathematical Monthly
1525:In the Renaissance theory of
1442:and each such pair defines a
66:(or, equivalently, powers of
3134:Journal of Cuneiform Studies
2446:Journal of Cuneiform Studies
2292:note that this follows from
2256:to organize 5-limit pitches.
5280:Multiplicative digital root
3342:10.1080/0749446032000134715
2773:Habens, Rev. W. J. (1889),
2659:A Discipline of Programming
2314:Howard & Longair (1982)
1981:-dimensional extremal even
1785:Python programming language
1365:. This scale is called a 5-
771:Even more precisely, using
246:, for nonnegative integers
129:regular sexagesimal numbers
93:, these numbers are called
5802:
3210:Sloane, N. J. A.
3031:. A correction appears in
2905:10.1016/j.jcta.2006.03.018
2686:Hamming's exercise in SASL
2662:, Prentice-Hall, pp.
2290:Halsey & Hewitt (1972)
2184:Fowler & Robson (1998)
2123:Berndt & Rankin (1995)
2005:th power of a polynomial.
31:
5740:
5723:
5709:
5687:
5673:
5651:
5637:
5615:
5601:
5574:
5561:
5357:Perfect digital invariant
5200:
5186:
5094:
5075:
4932:Superior highly composite
4818:
4801:
4729:
4716:
4684:
4671:
4559:
4548:
4251:
4238:
4196:
4185:
4148:
4137:
4085:
4071:
4034:
4023:
3976:
3961:
3879:
3865:
3672:Superior highly composite
3481:
3330:Contemporary Music Review
3007:Communications of the ACM
2944:10.1080/09298210500280612
2490:Asmussen, Robert (2001),
2008:As with other classes of
1369:tuning, meaning that the
1191:has been interpreted by
493:if and only if the point
4970:Euler's totient function
4754:Euler–Jacobi pseudoprime
4029:Other polynomial numbers
3569:Constrained divisor sums
3242:et leurs applications",
3066:10.1525/mp.2003.20.4.383
2230:Honingh & Bod (2005)
1871:{\displaystyle 60^{k+1}}
1668:Therefore, the sequence
1378:Honingh & Bod (2005)
1183:numbers from 56 to 80.)
1111:{\displaystyle 60^{k}/n}
688:can be estimated as the
32:Not to be confused with
4784:Somer–Lucas pseudoprime
4774:Lucas–Carmichael number
4609:Lazy caterer's sequence
3091:Journal of Music Theory
2634:, Copernicus, pp.
2197:Conway & Guy (1996)
1816:#Babylonian mathematics
1516:just chromatic semitone
1435:{\displaystyle (x,x+1)}
976:in his first letter to
524:{\displaystyle (i,j,k)}
5781:Functional programming
5771:Babylonian mathematics
4659:Wedderburn–Etherington
4059:Lucky numbers of Euler
3073:Longuet-Higgins, H. C.
2775:"On the musical scale"
2734:10.1006/hmat.1998.2209
2250:Longuet-Higgins (1962)
2014:fast Fourier transform
1999:
1975:
1955:
1924:
1900:
1872:
1839:
1838:{\displaystyle 60^{k}}
1808:
1755:
1732:
1709:
1682:
1665:is any Hamming number.
1659:
1639:
1616:
1593:
1539:harmonic whole numbers
1512:just diatonic semitone
1476:
1436:
1318:
1298:
1278:
1160:
1132:
1112:
1077:
1049:
1048:{\displaystyle 60^{k}}
1022:
997:
984:Babylonian mathematics
966:
946:
905:
789:
765:
682:
662:
603:
525:
487:
467:
363:
300:
280:
260:
240:
178:functional programming
155:harmonic whole numbers
121:Babylonian mathematics
56:
4947:Prime omega functions
4764:Frobenius pseudoprime
4554:Combinatorial numbers
4423:Centered dodecahedral
4216:Primary pseudoperfect
3450:Integer factorization
3372:10.1145/142137.142151
3020:10.1145/361454.361514
2000:
1976:
1956:
1925:
1901:
1873:
1840:
1809:
1756:
1733:
1710:
1683:
1660:
1640:
1617:
1594:
1477:
1444:superparticular ratio
1437:
1363:fundamental frequency
1319:
1299:
1279:
1161:
1133:
1113:
1078:
1050:
1023:
991:
967:
947:
906:
790:
766:
683:
663:
604:
526:
488:
468:
364:
301:
281:
261:
241:
42:
5786:Mathematics of music
5406:-composition related
5206:Arithmetic functions
4808:Arithmetic functions
4744:Elliptic pseudoprime
4428:Centered icosahedral
4408:Centered tetrahedral
2721:Historia Mathematica
1989:
1965:
1945:
1914:
1890:
1849:
1822:
1798:
1742:
1719:
1696:
1672:
1649:
1626:
1603:
1580:
1449:
1408:
1308:
1288:
1202:
1142:
1122:
1087:
1059:
1032:
1012:
956:
915:
799:
779:
696:
672:
613:
539:
497:
477:
418:
310:
290:
270:
250:
197:
5332:Kaprekar's constant
4852:Colossally abundant
4739:Catalan pseudoprime
4639:Schröder–Hipparchus
4418:Centered octahedral
4294:Centered heptagonal
4284:Centered pentagonal
4274:Centered triangular
3874:and related numbers
3662:Colossally abundant
3493:Factorization forms
3359:ACM SIGPLAN Notices
3306:2015EcolL..18..653V
2853:10.1145/44326.44335
2840:ACM SIGPLAN Notices
2681:Dijkstra, Edsger W.
2654:Dijkstra, Edsger W.
2631:The Book of Numbers
2604:1877Natur..15..253C
2563:1995rlc..book.....B
2501:University of Leeds
2327:Hemmendinger (1988)
2044:Certain species of
1197:Pythagorean triples
1159:{\displaystyle 1/n}
1076:{\displaystyle 1/n}
974:Srinivasa Ramanujan
5750:Mathematics portal
5692:Aronson's sequence
5438:Smarandache–Wellin
5195:-dependent numbers
4902:Primitive abundant
4789:Strong pseudoprime
4779:Perrin pseudoprime
4759:Fermat pseudoprime
4699:Wolstenholme prime
4523:Squared triangular
4309:Centered decagonal
4304:Centered nonagonal
4299:Centered octagonal
4289:Centered hexagonal
3647:Primitive abundant
3635:With many divisors
3086:McClain, Ernest G.
2344:test_generators.py
1995:
1983:unimodular lattice
1971:
1951:
1934:Other applications
1920:
1896:
1868:
1835:
1804:
1754:{\displaystyle 5H}
1751:
1731:{\displaystyle 3H}
1728:
1708:{\displaystyle 2H}
1705:
1678:
1655:
1638:{\displaystyle 5h}
1635:
1615:{\displaystyle 3h}
1612:
1592:{\displaystyle 2h}
1589:
1518:), and 81/80 (the
1472:
1470:
1432:
1314:
1294:
1274:
1156:
1128:
1108:
1073:
1045:
1018:
1000:In the Babylonian
998:
962:
942:
901:
785:
761:
678:
658:
599:
521:
483:
463:
359:
296:
276:
256:
236:
57:
5776:Integer sequences
5758:
5757:
5736:
5735:
5705:
5704:
5669:
5668:
5633:
5632:
5597:
5596:
5557:
5556:
5553:
5552:
5372:
5371:
5182:
5181:
5071:
5070:
5067:
5066:
5013:Aliquot sequences
4824:Divisor functions
4797:
4796:
4769:Lucas pseudoprime
4749:Euler pseudoprime
4734:Carmichael number
4712:
4711:
4667:
4666:
4544:
4543:
4540:
4539:
4536:
4535:
4497:
4496:
4385:
4384:
4342:Square triangular
4234:
4233:
4181:
4180:
4133:
4132:
4067:
4066:
4019:
4018:
3957:
3956:
3824:
3823:
3314:10.1111/ele.12442
3223:, OEIS Foundation
3079:(August): 244–248
2572:978-0-8218-0470-4
2342:Function m235 in
2294:Størmer's theorem
2018:time-varying data
1998:{\displaystyle n}
1974:{\displaystyle n}
1954:{\displaystyle n}
1923:{\displaystyle k}
1899:{\displaystyle k}
1807:{\displaystyle k}
1794:, is to list all
1681:{\displaystyle H}
1658:{\displaystyle h}
1563:Hamming's problem
1553:. Dijkstra (
1527:universal harmony
1469:
1399:equal temperament
1317:{\displaystyle q}
1297:{\displaystyle p}
1131:{\displaystyle n}
1083:is just that for
1021:{\displaystyle n}
965:{\displaystyle N}
875:
827:
788:{\displaystyle N}
756:
681:{\displaystyle N}
486:{\displaystyle N}
299:{\displaystyle k}
279:{\displaystyle j}
259:{\displaystyle i}
16:(Redirected from
5793:
5748:
5711:
5680:Natural language
5675:
5639:
5607:Generated via a
5603:
5563:
5468:Digit-reassembly
5433:Self-descriptive
5237:
5202:
5188:
5139:Lucas–Carmichael
5129:Harmonic divisor
5077:
5003:Sparsely totient
4978:Highly cototient
4887:Multiply perfect
4877:Highly composite
4820:
4803:
4718:
4673:
4654:Telephone number
4550:
4508:
4489:Square pyramidal
4471:Stella octangula
4396:
4262:
4253:
4245:Figurate numbers
4240:
4187:
4139:
4073:
4025:
3963:
3867:
3851:
3844:
3837:
3828:
3801:Harmonic divisor
3687:Aliquot sequence
3667:Highly composite
3591:Multiply perfect
3487:
3465:Divisor function
3431:
3424:
3417:
3408:
3382:
3352:
3324:
3287:
3262:
3251:
3241:
3224:
3204:
3173:
3126:
3106:
3080:
3068:
3054:Music Perception
3030:
3003:
2990:
2954:
2923:
2898:
2889:(8): 1732–1745,
2878:
2873:Sloane, N. J. A.
2871:; Rains, E. M.;
2863:
2832:
2793:
2768:
2737:
2736:
2718:
2707:
2693:
2691:
2676:
2648:
2616:
2615:
2613:10.1038/015253b0
2585:
2575:
2548:
2517:
2516:
2515:
2509:
2503:, archived from
2499:, Ph.D. thesis,
2498:
2485:
2426:
2420:
2411:
2401:
2395:
2392:Temperton (1992)
2389:
2383:
2377:
2371:
2365:
2359:
2356:Gingerich (1965)
2353:
2347:
2340:
2334:
2323:
2317:
2311:
2305:
2287:
2281:
2275:
2269:
2263:
2257:
2239:
2233:
2227:
2216:
2210:
2204:
2193:
2187:
2181:
2172:
2166:
2155:
2149:
2143:
2137:
2126:
2120:
2114:
2108:
2102:
2099:Pomerance (1995)
2096:
2090:
2087:Sloane "A051037"
2084:
2073:
2066:
2022:Temperton (1992)
2004:
2002:
2001:
1996:
1980:
1978:
1977:
1972:
1960:
1958:
1957:
1952:
1941:show that, when
1929:
1927:
1926:
1921:
1905:
1903:
1902:
1897:
1880:Gingerich (1965)
1877:
1875:
1874:
1869:
1867:
1866:
1844:
1842:
1841:
1836:
1834:
1833:
1813:
1811:
1810:
1805:
1760:
1758:
1757:
1752:
1737:
1735:
1734:
1729:
1714:
1712:
1711:
1706:
1687:
1685:
1684:
1679:
1664:
1662:
1661:
1656:
1644:
1642:
1641:
1636:
1621:
1619:
1618:
1613:
1598:
1596:
1595:
1590:
1500:just minor third
1496:just major third
1481:
1479:
1478:
1473:
1471:
1465:
1454:
1441:
1439:
1438:
1433:
1326:square root of 2
1323:
1321:
1320:
1315:
1303:
1301:
1300:
1295:
1283:
1281:
1280:
1275:
1270:
1269:
1257:
1256:
1230:
1229:
1217:
1216:
1165:
1163:
1162:
1157:
1152:
1137:
1135:
1134:
1129:
1117:
1115:
1114:
1109:
1104:
1099:
1098:
1082:
1080:
1079:
1074:
1069:
1054:
1052:
1051:
1046:
1044:
1043:
1027:
1025:
1024:
1019:
971:
969:
968:
963:
951:
949:
948:
943:
910:
908:
907:
902:
876:
874:
842:
841:
836:
832:
828:
823:
803:
794:
792:
791:
786:
770:
768:
767:
762:
757:
752:
745:
744:
728:
727:
711:
710:
700:
687:
685:
684:
679:
667:
665:
664:
659:
651:
650:
638:
637:
625:
624:
608:
606:
605:
600:
530:
528:
527:
522:
492:
490:
489:
484:
472:
470:
469:
464:
462:
461:
449:
448:
436:
435:
399:
388:
386:
380:
368:
366:
365:
360:
358:
357:
335:
305:
303:
302:
297:
285:
283:
282:
277:
265:
263:
262:
257:
245:
243:
242:
237:
235:
234:
222:
221:
209:
208:
162:computer science
119:In the study of
116:
114:
104:
21:
5801:
5800:
5796:
5795:
5794:
5792:
5791:
5790:
5761:
5760:
5759:
5754:
5732:
5728:Strobogrammatic
5719:
5701:
5683:
5665:
5647:
5629:
5611:
5593:
5570:
5549:
5533:
5492:Divisor-related
5487:
5447:
5398:
5368:
5305:
5289:
5268:
5235:
5208:
5196:
5178:
5090:
5089:related numbers
5063:
5040:
5007:
4998:Perfect totient
4964:
4941:
4872:Highly abundant
4814:
4793:
4725:
4708:
4680:
4663:
4649:Stirling second
4555:
4532:
4493:
4475:
4432:
4381:
4318:
4279:Centered square
4247:
4230:
4192:
4177:
4144:
4129:
4081:
4080:defined numbers
4063:
4030:
4015:
3986:Double Mersenne
3972:
3953:
3875:
3861:
3859:natural numbers
3855:
3825:
3820:
3764:
3720:
3681:
3652:Highly abundant
3630:
3611:Unitary perfect
3564:
3488:
3479:
3460:Unitary divisor
3438:
3435:
3391:
3386:
3355:
3327:
3294:Ecology Letters
3291:
3277:10.1137/0913039
3257:
3255:
3232:
3227:
3208:
3194:10.2307/2316896
3177:
3147:10.2307/1359434
3130:
3112:Pomerance, Carl
3110:
3084:
3071:
3051:
3001:
2993:
2960:Howard, Deborah
2958:
2927:
2896:math.NT/0509316
2876:
2869:Heninger, Nadia
2867:
2836:
2814:10.2307/2317424
2797:
2772:
2758:10.2307/1006080
2741:
2716:
2711:
2699:Eppstein, David
2697:
2689:
2679:
2674:
2652:
2646:
2626:Guy, Richard K.
2622:Conway, John H.
2620:
2589:
2579:
2573:
2552:
2521:
2513:
2511:
2507:
2496:
2489:
2459:10.2307/1359089
2439:
2435:
2430:
2429:
2421:
2414:
2402:
2398:
2390:
2386:
2378:
2374:
2368:Eppstein (2007)
2366:
2362:
2354:
2350:
2341:
2337:
2324:
2320:
2312:
2308:
2288:
2284:
2276:
2272:
2264:
2260:
2242:Asmussen (2001)
2240:
2236:
2228:
2219:
2211:
2207:
2194:
2190:
2182:
2175:
2167:
2158:
2150:
2146:
2138:
2129:
2121:
2117:
2109:
2105:
2097:
2093:
2085:
2076:
2067:
2063:
2058:
1987:
1986:
1963:
1962:
1943:
1942:
1936:
1912:
1911:
1908:Eppstein (2007)
1888:
1887:
1852:
1847:
1846:
1825:
1820:
1819:
1796:
1795:
1740:
1739:
1717:
1716:
1694:
1693:
1670:
1669:
1647:
1646:
1624:
1623:
1601:
1600:
1578:
1577:
1567:Hamming numbers
1551:Edsger Dijkstra
1547:
1508:just minor tone
1504:just major tone
1455:
1447:
1446:
1406:
1405:
1342:just intonation
1334:
1306:
1305:
1286:
1285:
1261:
1248:
1221:
1208:
1200:
1199:
1140:
1139:
1120:
1119:
1090:
1085:
1084:
1057:
1056:
1035:
1030:
1029:
1010:
1009:
986:
954:
953:
913:
912:
843:
809:
805:
804:
797:
796:
777:
776:
736:
719:
702:
701:
694:
693:
670:
669:
642:
629:
616:
611:
610:
537:
536:
531:belongs to the
495:
494:
475:
474:
453:
440:
427:
416:
415:
405:
395:
384:
382:
378:
313:
308:
307:
288:
287:
268:
267:
248:
247:
226:
213:
200:
195:
194:
187:
170:Richard Hamming
166:Hamming numbers
147:just intonation
125:regular numbers
112:
110:
102:
60:Regular numbers
37:
30:
23:
22:
18:Regular numbers
15:
12:
11:
5:
5799:
5797:
5789:
5788:
5783:
5778:
5773:
5763:
5762:
5756:
5755:
5753:
5752:
5741:
5738:
5737:
5734:
5733:
5731:
5730:
5724:
5721:
5720:
5714:
5707:
5706:
5703:
5702:
5700:
5699:
5694:
5688:
5685:
5684:
5678:
5671:
5670:
5667:
5666:
5664:
5663:
5661:Sorting number
5658:
5656:Pancake number
5652:
5649:
5648:
5642:
5635:
5634:
5631:
5630:
5628:
5627:
5622:
5616:
5613:
5612:
5606:
5599:
5598:
5595:
5594:
5592:
5591:
5586:
5581:
5575:
5572:
5571:
5568:Binary numbers
5566:
5559:
5558:
5555:
5554:
5551:
5550:
5548:
5547:
5541:
5539:
5535:
5534:
5532:
5531:
5526:
5521:
5516:
5511:
5506:
5501:
5495:
5493:
5489:
5488:
5486:
5485:
5480:
5475:
5470:
5465:
5459:
5457:
5449:
5448:
5446:
5445:
5440:
5435:
5430:
5425:
5420:
5415:
5409:
5407:
5400:
5399:
5397:
5396:
5395:
5394:
5383:
5381:
5378:P-adic numbers
5374:
5373:
5370:
5369:
5367:
5366:
5365:
5364:
5354:
5349:
5344:
5339:
5334:
5329:
5324:
5319:
5313:
5311:
5307:
5306:
5304:
5303:
5297:
5295:
5294:Coding-related
5291:
5290:
5288:
5287:
5282:
5276:
5274:
5270:
5269:
5267:
5266:
5261:
5256:
5251:
5245:
5243:
5234:
5233:
5232:
5231:
5229:Multiplicative
5226:
5215:
5213:
5198:
5197:
5193:Numeral system
5191:
5184:
5183:
5180:
5179:
5177:
5176:
5171:
5166:
5161:
5156:
5151:
5146:
5141:
5136:
5131:
5126:
5121:
5116:
5111:
5106:
5101:
5095:
5092:
5091:
5080:
5073:
5072:
5069:
5068:
5065:
5064:
5062:
5061:
5056:
5050:
5048:
5042:
5041:
5039:
5038:
5033:
5028:
5023:
5017:
5015:
5009:
5008:
5006:
5005:
5000:
4995:
4990:
4985:
4983:Highly totient
4980:
4974:
4972:
4966:
4965:
4963:
4962:
4957:
4951:
4949:
4943:
4942:
4940:
4939:
4934:
4929:
4924:
4919:
4914:
4909:
4904:
4899:
4894:
4889:
4884:
4879:
4874:
4869:
4864:
4859:
4854:
4849:
4844:
4839:
4837:Almost perfect
4834:
4828:
4826:
4816:
4815:
4806:
4799:
4798:
4795:
4794:
4792:
4791:
4786:
4781:
4776:
4771:
4766:
4761:
4756:
4751:
4746:
4741:
4736:
4730:
4727:
4726:
4721:
4714:
4713:
4710:
4709:
4707:
4706:
4701:
4696:
4691:
4685:
4682:
4681:
4676:
4669:
4668:
4665:
4664:
4662:
4661:
4656:
4651:
4646:
4644:Stirling first
4641:
4636:
4631:
4626:
4621:
4616:
4611:
4606:
4601:
4596:
4591:
4586:
4581:
4576:
4571:
4566:
4560:
4557:
4556:
4553:
4546:
4545:
4542:
4541:
4538:
4537:
4534:
4533:
4531:
4530:
4525:
4520:
4514:
4512:
4505:
4499:
4498:
4495:
4494:
4492:
4491:
4485:
4483:
4477:
4476:
4474:
4473:
4468:
4463:
4458:
4453:
4448:
4442:
4440:
4434:
4433:
4431:
4430:
4425:
4420:
4415:
4410:
4404:
4402:
4393:
4387:
4386:
4383:
4382:
4380:
4379:
4374:
4369:
4364:
4359:
4354:
4349:
4344:
4339:
4334:
4328:
4326:
4320:
4319:
4317:
4316:
4311:
4306:
4301:
4296:
4291:
4286:
4281:
4276:
4270:
4268:
4259:
4249:
4248:
4243:
4236:
4235:
4232:
4231:
4229:
4228:
4223:
4218:
4213:
4208:
4203:
4197:
4194:
4193:
4190:
4183:
4182:
4179:
4178:
4176:
4175:
4170:
4165:
4160:
4155:
4149:
4146:
4145:
4142:
4135:
4134:
4131:
4130:
4128:
4127:
4122:
4117:
4112:
4107:
4102:
4097:
4092:
4086:
4083:
4082:
4076:
4069:
4068:
4065:
4064:
4062:
4061:
4056:
4051:
4046:
4041:
4035:
4032:
4031:
4028:
4021:
4020:
4017:
4016:
4014:
4013:
4008:
4003:
3998:
3993:
3988:
3983:
3977:
3974:
3973:
3966:
3959:
3958:
3955:
3954:
3952:
3951:
3946:
3941:
3936:
3931:
3926:
3921:
3916:
3911:
3906:
3901:
3896:
3891:
3886:
3880:
3877:
3876:
3870:
3863:
3862:
3856:
3854:
3853:
3846:
3839:
3831:
3822:
3821:
3819:
3818:
3813:
3808:
3803:
3798:
3793:
3788:
3783:
3778:
3772:
3770:
3766:
3765:
3763:
3762:
3757:
3752:
3747:
3742:
3737:
3731:
3729:
3722:
3721:
3719:
3718:
3713:
3708:
3698:
3692:
3690:
3683:
3682:
3680:
3679:
3674:
3669:
3664:
3659:
3654:
3649:
3644:
3638:
3636:
3632:
3631:
3629:
3628:
3623:
3618:
3613:
3608:
3603:
3598:
3593:
3588:
3583:
3581:Almost perfect
3578:
3572:
3570:
3566:
3565:
3563:
3562:
3557:
3552:
3547:
3542:
3537:
3532:
3527:
3522:
3517:
3512:
3507:
3502:
3496:
3494:
3490:
3489:
3482:
3480:
3478:
3477:
3472:
3467:
3462:
3457:
3452:
3446:
3444:
3440:
3439:
3436:
3434:
3433:
3426:
3419:
3411:
3405:
3404:
3398:
3390:
3389:External links
3387:
3385:
3384:
3353:
3325:
3300:(7): 653–659,
3289:
3271:(3): 676–686,
3253:
3225:
3206:
3175:
3128:
3108:
3082:
3069:
3060:(4): 383–410,
3049:
3014:(7): 671–677,
2991:
2980:10.2307/989675
2974:(2): 116–143,
2956:
2938:(3): 293–303,
2925:
2865:
2834:
2795:
2770:
2739:
2727:(4): 366–378,
2709:
2695:
2677:
2673:978-0132158718
2672:
2650:
2644:
2618:
2587:
2577:
2571:
2550:
2538:10.2307/592627
2519:
2487:
2436:
2434:
2431:
2428:
2427:
2412:
2408:McClain (1974)
2396:
2384:
2372:
2360:
2348:
2335:
2318:
2306:
2282:
2270:
2258:
2234:
2217:
2205:
2188:
2173:
2156:
2144:
2127:
2115:
2103:
2091:
2074:
2060:
2059:
2057:
2054:
2039:Plato's number
2010:smooth numbers
1994:
1970:
1950:
1935:
1932:
1919:
1895:
1865:
1862:
1859:
1855:
1832:
1828:
1803:
1763:
1762:
1750:
1747:
1727:
1724:
1704:
1701:
1692:the sequences
1677:
1666:
1654:
1634:
1631:
1611:
1608:
1588:
1585:
1574:
1546:
1543:
1520:syntonic comma
1514:), 25/24 (the
1510:), 16/15 (the
1492:perfect fourth
1468:
1464:
1461:
1458:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1397:. However the
1346:diatonic scale
1333:
1330:
1313:
1293:
1273:
1268:
1264:
1260:
1255:
1251:
1246:
1243:
1240:
1237:
1233:
1228:
1224:
1220:
1215:
1211:
1207:
1166:, then shift.
1155:
1151:
1147:
1138:, multiply by
1127:
1107:
1103:
1097:
1093:
1072:
1068:
1064:
1042:
1038:
1017:
1004:notation, the
985:
982:
961:
941:
938:
935:
932:
929:
926:
923:
920:
900:
897:
894:
891:
888:
885:
882:
879:
873:
870:
867:
864:
861:
858:
855:
852:
849:
846:
840:
835:
831:
826:
821:
818:
815:
812:
808:
784:
773:big O notation
760:
755:
751:
748:
743:
739:
734:
731:
726:
722:
717:
714:
709:
705:
677:
657:
654:
649:
645:
641:
636:
632:
628:
623:
619:
598:
595:
592:
589:
586:
583:
580:
577:
574:
571:
568:
565:
562:
559:
556:
553:
550:
547:
544:
520:
517:
514:
511:
508:
505:
502:
482:
460:
456:
452:
447:
443:
439:
434:
430:
426:
423:
393:
356:
353:
350:
347:
344:
341:
338:
334:
329:
326:
323:
320:
316:
295:
275:
255:
233:
229:
225:
220:
216:
212:
207:
203:
186:
183:
182:
181:
158:
136:
117:
107:smooth numbers
28:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5798:
5787:
5784:
5782:
5779:
5777:
5774:
5772:
5769:
5768:
5766:
5751:
5747:
5743:
5742:
5739:
5729:
5726:
5725:
5722:
5717:
5712:
5708:
5698:
5695:
5693:
5690:
5689:
5686:
5681:
5676:
5672:
5662:
5659:
5657:
5654:
5653:
5650:
5645:
5640:
5636:
5626:
5623:
5621:
5618:
5617:
5614:
5610:
5604:
5600:
5590:
5587:
5585:
5582:
5580:
5577:
5576:
5573:
5569:
5564:
5560:
5546:
5543:
5542:
5540:
5536:
5530:
5527:
5525:
5522:
5520:
5519:Polydivisible
5517:
5515:
5512:
5510:
5507:
5505:
5502:
5500:
5497:
5496:
5494:
5490:
5484:
5481:
5479:
5476:
5474:
5471:
5469:
5466:
5464:
5461:
5460:
5458:
5455:
5450:
5444:
5441:
5439:
5436:
5434:
5431:
5429:
5426:
5424:
5421:
5419:
5416:
5414:
5411:
5410:
5408:
5405:
5401:
5393:
5390:
5389:
5388:
5385:
5384:
5382:
5379:
5375:
5363:
5360:
5359:
5358:
5355:
5353:
5350:
5348:
5345:
5343:
5340:
5338:
5335:
5333:
5330:
5328:
5325:
5323:
5320:
5318:
5315:
5314:
5312:
5308:
5302:
5299:
5298:
5296:
5292:
5286:
5283:
5281:
5278:
5277:
5275:
5273:Digit product
5271:
5265:
5262:
5260:
5257:
5255:
5252:
5250:
5247:
5246:
5244:
5242:
5238:
5230:
5227:
5225:
5222:
5221:
5220:
5217:
5216:
5214:
5212:
5207:
5203:
5199:
5194:
5189:
5185:
5175:
5172:
5170:
5167:
5165:
5162:
5160:
5157:
5155:
5152:
5150:
5147:
5145:
5142:
5140:
5137:
5135:
5132:
5130:
5127:
5125:
5122:
5120:
5117:
5115:
5112:
5110:
5109:Erdős–Nicolas
5107:
5105:
5102:
5100:
5097:
5096:
5093:
5088:
5084:
5078:
5074:
5060:
5057:
5055:
5052:
5051:
5049:
5047:
5043:
5037:
5034:
5032:
5029:
5027:
5024:
5022:
5019:
5018:
5016:
5014:
5010:
5004:
5001:
4999:
4996:
4994:
4991:
4989:
4986:
4984:
4981:
4979:
4976:
4975:
4973:
4971:
4967:
4961:
4958:
4956:
4953:
4952:
4950:
4948:
4944:
4938:
4935:
4933:
4930:
4928:
4927:Superabundant
4925:
4923:
4920:
4918:
4915:
4913:
4910:
4908:
4905:
4903:
4900:
4898:
4895:
4893:
4890:
4888:
4885:
4883:
4880:
4878:
4875:
4873:
4870:
4868:
4865:
4863:
4860:
4858:
4855:
4853:
4850:
4848:
4845:
4843:
4840:
4838:
4835:
4833:
4830:
4829:
4827:
4825:
4821:
4817:
4813:
4809:
4804:
4800:
4790:
4787:
4785:
4782:
4780:
4777:
4775:
4772:
4770:
4767:
4765:
4762:
4760:
4757:
4755:
4752:
4750:
4747:
4745:
4742:
4740:
4737:
4735:
4732:
4731:
4728:
4724:
4719:
4715:
4705:
4702:
4700:
4697:
4695:
4692:
4690:
4687:
4686:
4683:
4679:
4674:
4670:
4660:
4657:
4655:
4652:
4650:
4647:
4645:
4642:
4640:
4637:
4635:
4632:
4630:
4627:
4625:
4622:
4620:
4617:
4615:
4612:
4610:
4607:
4605:
4602:
4600:
4597:
4595:
4592:
4590:
4587:
4585:
4582:
4580:
4577:
4575:
4572:
4570:
4567:
4565:
4562:
4561:
4558:
4551:
4547:
4529:
4526:
4524:
4521:
4519:
4516:
4515:
4513:
4509:
4506:
4504:
4503:4-dimensional
4500:
4490:
4487:
4486:
4484:
4482:
4478:
4472:
4469:
4467:
4464:
4462:
4459:
4457:
4454:
4452:
4449:
4447:
4444:
4443:
4441:
4439:
4435:
4429:
4426:
4424:
4421:
4419:
4416:
4414:
4413:Centered cube
4411:
4409:
4406:
4405:
4403:
4401:
4397:
4394:
4392:
4391:3-dimensional
4388:
4378:
4375:
4373:
4370:
4368:
4365:
4363:
4360:
4358:
4355:
4353:
4350:
4348:
4345:
4343:
4340:
4338:
4335:
4333:
4330:
4329:
4327:
4325:
4321:
4315:
4312:
4310:
4307:
4305:
4302:
4300:
4297:
4295:
4292:
4290:
4287:
4285:
4282:
4280:
4277:
4275:
4272:
4271:
4269:
4267:
4263:
4260:
4258:
4257:2-dimensional
4254:
4250:
4246:
4241:
4237:
4227:
4224:
4222:
4219:
4217:
4214:
4212:
4209:
4207:
4204:
4202:
4201:Nonhypotenuse
4199:
4198:
4195:
4188:
4184:
4174:
4171:
4169:
4166:
4164:
4161:
4159:
4156:
4154:
4151:
4150:
4147:
4140:
4136:
4126:
4123:
4121:
4118:
4116:
4113:
4111:
4108:
4106:
4103:
4101:
4098:
4096:
4093:
4091:
4088:
4087:
4084:
4079:
4074:
4070:
4060:
4057:
4055:
4052:
4050:
4047:
4045:
4042:
4040:
4037:
4036:
4033:
4026:
4022:
4012:
4009:
4007:
4004:
4002:
3999:
3997:
3994:
3992:
3989:
3987:
3984:
3982:
3979:
3978:
3975:
3970:
3964:
3960:
3950:
3947:
3945:
3942:
3940:
3939:Perfect power
3937:
3935:
3932:
3930:
3929:Seventh power
3927:
3925:
3922:
3920:
3917:
3915:
3912:
3910:
3907:
3905:
3902:
3900:
3897:
3895:
3892:
3890:
3887:
3885:
3882:
3881:
3878:
3873:
3868:
3864:
3860:
3852:
3847:
3845:
3840:
3838:
3833:
3832:
3829:
3817:
3814:
3812:
3809:
3807:
3804:
3802:
3799:
3797:
3794:
3792:
3789:
3787:
3784:
3782:
3779:
3777:
3774:
3773:
3771:
3767:
3761:
3758:
3756:
3755:Polydivisible
3753:
3751:
3748:
3746:
3743:
3741:
3738:
3736:
3733:
3732:
3730:
3727:
3723:
3717:
3714:
3712:
3709:
3706:
3702:
3699:
3697:
3694:
3693:
3691:
3688:
3684:
3678:
3675:
3673:
3670:
3668:
3665:
3663:
3660:
3658:
3657:Superabundant
3655:
3653:
3650:
3648:
3645:
3643:
3640:
3639:
3637:
3633:
3627:
3626:Erdős–Nicolas
3624:
3622:
3619:
3617:
3614:
3612:
3609:
3607:
3604:
3602:
3599:
3597:
3594:
3592:
3589:
3587:
3584:
3582:
3579:
3577:
3574:
3573:
3571:
3567:
3561:
3558:
3556:
3553:
3551:
3548:
3546:
3543:
3541:
3538:
3536:
3535:Perfect power
3533:
3531:
3528:
3526:
3523:
3521:
3518:
3516:
3513:
3511:
3508:
3506:
3503:
3501:
3498:
3497:
3495:
3491:
3486:
3476:
3473:
3471:
3468:
3466:
3463:
3461:
3458:
3456:
3453:
3451:
3448:
3447:
3445:
3441:
3432:
3427:
3425:
3420:
3418:
3413:
3412:
3409:
3402:
3399:
3396:
3393:
3392:
3388:
3381:
3377:
3373:
3369:
3365:
3361:
3360:
3354:
3351:
3347:
3343:
3339:
3336:(1–2): 3–14,
3335:
3331:
3326:
3323:
3319:
3315:
3311:
3307:
3303:
3299:
3295:
3290:
3286:
3282:
3278:
3274:
3270:
3266:
3260:
3254:
3249:
3245:
3239:
3235:
3230:
3229:Størmer, Carl
3226:
3222:
3221:
3215:
3211:
3207:
3203:
3199:
3195:
3191:
3187:
3183:
3182:
3176:
3172:
3168:
3164:
3160:
3156:
3152:
3148:
3144:
3140:
3136:
3135:
3129:
3125:
3121:
3117:
3113:
3109:
3105:
3101:
3097:
3093:
3092:
3087:
3083:
3078:
3074:
3070:
3067:
3063:
3059:
3055:
3050:
3047:
3045:
3040:
3038:
3035:
3029:
3025:
3021:
3017:
3013:
3009:
3008:
3000:
2996:
2992:
2989:
2985:
2981:
2977:
2973:
2969:
2965:
2964:Quattro Libri
2961:
2957:
2953:
2949:
2945:
2941:
2937:
2933:
2932:
2926:
2922:
2918:
2914:
2910:
2906:
2902:
2897:
2892:
2888:
2884:
2883:
2874:
2870:
2866:
2862:
2858:
2854:
2850:
2846:
2842:
2841:
2835:
2831:
2827:
2823:
2819:
2815:
2811:
2807:
2803:
2802:
2796:
2792:
2788:
2784:
2780:
2776:
2771:
2767:
2763:
2759:
2755:
2751:
2747:
2746:
2740:
2735:
2730:
2726:
2722:
2715:
2710:
2706:
2705:
2700:
2696:
2688:
2687:
2682:
2678:
2675:
2669:
2665:
2661:
2660:
2655:
2651:
2647:
2645:0-387-97993-X
2641:
2637:
2633:
2632:
2627:
2623:
2619:
2614:
2609:
2605:
2601:
2597:
2593:
2588:
2583:
2578:
2574:
2568:
2564:
2560:
2556:
2551:
2547:
2543:
2539:
2535:
2531:
2527:
2526:
2520:
2510:on 2016-04-24
2506:
2502:
2495:
2494:
2488:
2484:
2480:
2476:
2472:
2468:
2464:
2460:
2456:
2452:
2448:
2447:
2442:
2438:
2437:
2432:
2424:
2419:
2417:
2413:
2409:
2405:
2404:Barton (1908)
2400:
2397:
2393:
2388:
2385:
2381:
2376:
2373:
2369:
2364:
2361:
2357:
2352:
2349:
2345:
2339:
2336:
2332:
2328:
2322:
2319:
2315:
2310:
2307:
2303:
2302:Silver (1971)
2299:
2295:
2291:
2286:
2283:
2279:
2274:
2271:
2267:
2266:Kopiez (2003)
2262:
2259:
2255:
2251:
2247:
2246:Habens (1889)
2243:
2238:
2235:
2231:
2226:
2224:
2222:
2218:
2214:
2213:Clarke (1877)
2209:
2206:
2202:
2198:
2192:
2189:
2185:
2180:
2178:
2174:
2170:
2165:
2163:
2161:
2157:
2153:
2148:
2145:
2141:
2136:
2134:
2132:
2128:
2124:
2119:
2116:
2112:
2107:
2104:
2100:
2095:
2092:
2088:
2083:
2081:
2079:
2075:
2071:
2065:
2062:
2055:
2053:
2051:
2047:
2042:
2040:
2036:
2035:
2030:
2027:Book VIII of
2025:
2023:
2019:
2015:
2011:
2006:
1992:
1984:
1968:
1948:
1940:
1933:
1931:
1917:
1909:
1893:
1885:
1884:Bruins (1970)
1881:
1863:
1860:
1857:
1853:
1830:
1826:
1817:
1801:
1793:
1788:
1786:
1781:
1779:
1775:
1771:
1768:
1748:
1745:
1725:
1722:
1702:
1699:
1691:
1675:
1667:
1652:
1632:
1629:
1609:
1606:
1586:
1583:
1575:
1572:
1571:
1570:
1568:
1564:
1560:
1556:
1552:
1544:
1542:
1540:
1536:
1532:
1528:
1523:
1521:
1517:
1513:
1509:
1506:), 10/9 (the
1505:
1501:
1497:
1493:
1489:
1488:perfect fifth
1485:
1466:
1462:
1459:
1456:
1445:
1426:
1423:
1420:
1417:
1414:
1402:
1400:
1396:
1392:
1388:
1384:
1379:
1374:
1372:
1368:
1364:
1360:
1355:
1351:
1347:
1343:
1339:
1331:
1329:
1327:
1311:
1291:
1284:generated by
1266:
1262:
1258:
1253:
1249:
1244:
1241:
1238:
1235:
1231:
1226:
1222:
1218:
1213:
1209:
1198:
1194:
1190:
1184:
1181:
1176:
1171:
1167:
1153:
1149:
1145:
1125:
1105:
1101:
1095:
1091:
1070:
1066:
1062:
1040:
1036:
1015:
1007:
1003:
995:
994:Seleucid Uruk
990:
983:
981:
979:
975:
959:
936:
933:
930:
927:
924:
918:
898:
892:
889:
886:
880:
877:
871:
868:
865:
862:
859:
856:
853:
850:
847:
844:
838:
833:
824:
819:
813:
810:
806:
782:
774:
758:
753:
749:
746:
741:
737:
732:
729:
724:
720:
715:
712:
707:
703:
691:
675:
655:
652:
647:
643:
639:
634:
630:
626:
621:
617:
596:
593:
590:
587:
584:
581:
578:
575:
572:
569:
566:
563:
560:
557:
554:
551:
548:
545:
542:
534:
515:
512:
509:
506:
503:
480:
458:
454:
450:
445:
441:
437:
432:
428:
424:
421:
412:
410:
403:
398:
392:
389:
376:
372:
351:
348:
345:
342:
336:
332:
327:
314:
293:
273:
253:
231:
227:
223:
218:
214:
210:
205:
201:
192:
185:Number theory
184:
179:
175:
171:
167:
163:
159:
156:
152:
148:
145:
141:
137:
134:
130:
126:
122:
118:
108:
100:
99:prime factors
96:
92:
91:number theory
88:
87:
86:
83:
81:
77:
73:
69:
65:
61:
54:
50:
46:
45:Hasse diagram
41:
35:
34:regular prime
27:
19:
5483:Transposable
5347:Narcissistic
5254:Digital root
5174:Super-Poulet
5148:
5134:Jordan–Pólya
5083:prime factor
4988:Noncototient
4955:Almost prime
4937:Superperfect
4912:Refactorable
4907:Quasiperfect
4882:Hyperperfect
4723:Pseudoprimes
4694:Wall–Sun–Sun
4629:Ordered Bell
4599:Fuss–Catalan
4511:non-centered
4461:Dodecahedral
4438:non-centered
4324:non-centered
4226:Wolstenholme
3971:× 2 ± 1
3968:
3967:Of the form
3934:Eighth power
3914:Fourth power
3816:Superperfect
3811:Refactorable
3606:Superperfect
3601:Hyperperfect
3586:Quasiperfect
3549:
3470:Prime factor
3366:(8): 71–75,
3363:
3357:
3333:
3329:
3297:
3293:
3268:
3264:
3258:
3247:
3243:
3237:
3233:
3217:
3185:
3179:
3138:
3132:
3115:
3095:
3089:
3077:Music Review
3076:
3057:
3053:
3043:
3036:
3033:
3011:
3005:
2995:Knuth, D. E.
2971:
2967:
2963:
2935:
2929:
2886:
2885:, Series A,
2880:
2847:(4): 81–86,
2844:
2838:
2805:
2799:
2782:
2778:
2749:
2743:
2724:
2720:
2703:
2685:
2658:
2630:
2598:(377): 253,
2595:
2591:
2581:
2554:
2529:
2523:
2512:, retrieved
2505:the original
2492:
2450:
2444:
2441:Aaboe, Asger
2399:
2387:
2375:
2363:
2351:
2338:
2321:
2309:
2298:Størmer 1897
2285:
2273:
2261:
2237:
2208:
2201:Plimpton 322
2191:
2169:Knuth (1972)
2152:Sachs (1947)
2147:
2140:Aaboe (1965)
2118:
2106:
2094:
2064:
2043:
2032:
2026:
2007:
1937:
1792:Knuth (1972)
1789:
1782:
1764:
1566:
1562:
1548:
1538:
1531:architecture
1524:
1502:), 9/8 (the
1498:), 6/5 (the
1494:), 5/4 (the
1490:), 4/3 (the
1486:), 3/2 (the
1403:
1375:
1361:of a single
1352:in a single
1338:music theory
1335:
1332:Music theory
1189:Plimpton 322
1185:
1180:Donald Knuth
1172:
1168:
999:
972:is given by
413:
406:
390:
375:prime factor
193:of the form
188:
165:
154:
151:architecture
140:music theory
128:
124:
94:
84:
59:
58:
49:divisibility
26:
5504:Extravagant
5499:Equidigital
5454:permutation
5413:Palindromic
5387:Automorphic
5285:Sum-product
5264:Sum-product
5219:Persistence
5114:Erdős–Woods
5036:Untouchable
4917:Semiperfect
4867:Hemiperfect
4528:Tesseractic
4466:Icosahedral
4446:Tetrahedral
4377:Dodecagonal
4078:Recursively
3949:Prime power
3924:Sixth power
3919:Fifth power
3899:Power of 10
3857:Classes of
3740:Extravagant
3735:Equidigital
3696:Untouchable
3616:Semiperfect
3596:Hemiperfect
3525:Square-free
3401:RosettaCode
2738:, page 375.
2331:Yuen (1992)
2325:See, e.g.,
2278:Wolf (2003)
1195:as listing
1002:sexagesimal
978:G. H. Hardy
533:tetrahedron
133:sexagesimal
53:logarithmic
5765:Categories
5716:Graphemics
5589:Pernicious
5443:Undulating
5418:Pandigital
5392:Trimorphic
4993:Nontotient
4842:Arithmetic
4456:Octahedral
4357:Heptagonal
4347:Pentagonal
4332:Triangular
4173:Sierpiński
4095:Jacobsthal
3894:Power of 3
3889:Power of 2
3776:Arithmetic
3769:Other sets
3728:-dependent
2514:2007-03-15
2433:References
1778:generative
1774:imperative
1545:Algorithms
1193:Neugebauer
1006:reciprocal
174:algorithms
144:five-limit
5473:Parasitic
5322:Factorion
5249:Digit sum
5241:Digit sum
5059:Fortunate
5046:Primorial
4960:Semiprime
4897:Practical
4862:Descartes
4857:Deficient
4847:Betrothed
4689:Wieferich
4518:Pentatope
4481:pyramidal
4372:Decagonal
4367:Nonagonal
4362:Octagonal
4352:Hexagonal
4211:Practical
4158:Congruent
4090:Fibonacci
4054:Loeschian
3806:Descartes
3781:Deficient
3716:Betrothed
3621:Practical
3510:Semiprime
3505:Composite
3350:191457676
3171:163783242
3039:(2), 1976
2483:164195082
1395:consonant
1359:harmonics
1219:−
934:
928:
890:
869:
860:
851:
814:
747:
730:
713:
653:≤
640:⋅
627:⋅
591:
585:≤
579:
564:
549:
451:⋅
438:⋅
340:⌉
325:⌈
224:⋅
211:⋅
5545:Friedman
5478:Primeval
5423:Repdigit
5380:-related
5327:Kaprekar
5301:Meertens
5224:Additive
5211:dynamics
5119:Friendly
5031:Sociable
5021:Amicable
4832:Abundant
4812:dynamics
4634:Schröder
4624:Narayana
4594:Eulerian
4584:Delannoy
4579:Dedekind
4400:centered
4266:centered
4153:Amenable
4110:Narayana
4100:Leonardo
3996:Mersenne
3944:Powerful
3884:Achilles
3791:Solitary
3786:Friendly
3711:Sociable
3701:Amicable
3689:-related
3642:Abundant
3540:Achilles
3530:Powerful
3443:Overview
3380:18283005
3322:25963600
3285:14764894
3236:−
2997:(1972),
2952:16321292
2921:15913795
2861:28906392
2701:(2007),
2683:(1981),
2628:(1996),
2034:Republic
1645:, where
1535:Palladio
1371:interval
1175:Seleucid
1028:divides
168:, after
95:5-smooth
5718:related
5682:related
5646:related
5644:Sorting
5529:Vampire
5514:Harshad
5456:related
5428:Repunit
5342:Lychrel
5317:Dudeney
5169:Størmer
5164:Sphenic
5149:Regular
5087:divisor
5026:Perfect
4922:Sublime
4892:Perfect
4619:Motzkin
4574:Catalan
4115:Padovan
4049:Leyland
4044:Idoneal
4039:Hilbert
4011:Woodall
3796:Sublime
3750:Harshad
3576:Perfect
3560:Unusual
3550:Regular
3520:Sphenic
3455:Divisor
3302:Bibcode
3212:(ed.),
3202:2316896
3163:0022180
3155:1359434
3124:1403941
3028:7829945
2913:2269551
2830:0313189
2822:2317424
2766:1006080
2664:129–134
2636:172–176
2600:Bibcode
2559:Bibcode
2475:0191779
2467:1359089
2254:tonnetz
2050:masting
1783:In the
1690:merging
1387:tonnetz
1350:pitches
1344:of the
400:in the
397:A051037
191:integer
5584:Odious
5509:Frugal
5463:Cyclic
5452:Digit-
5159:Smooth
5144:Pronic
5104:Cyclic
5081:Other
5054:Euclid
4704:Wilson
4678:Primes
4337:Square
4206:Polite
4168:Riesel
4163:Knödel
4125:Perrin
4006:Thabit
3991:Fermat
3981:Cullen
3904:Square
3872:Powers
3745:Frugal
3705:Triple
3545:Smooth
3515:Pronic
3378:
3348:
3320:
3283:
3200:
3169:
3161:
3153:
3122:
3104:843638
3102:
3026:
2988:989675
2986:
2950:
2919:
2911:
2859:
2828:
2820:
2791:765355
2789:
2764:
2670:
2642:
2592:Nature
2569:
2546:592627
2544:
2481:
2473:
2465:
2046:bamboo
1985:is an
1878:. See
1738:, and
1622:, and
1484:octave
1354:octave
1340:, the
690:volume
371:smooth
286:, and
78:, and
5625:Prime
5620:Lucky
5609:sieve
5538:Other
5524:Smith
5404:Digit
5362:Happy
5337:Keith
5310:Other
5154:Rough
5124:Giuga
4589:Euler
4451:Cubic
4105:Lucas
4001:Proth
3760:Smith
3677:Weird
3555:Rough
3500:Prime
3376:S2CID
3346:S2CID
3281:S2CID
3261:= 235
3198:JSTOR
3167:S2CID
3151:JSTOR
3100:JSTOR
3024:S2CID
3002:(PDF)
2984:JSTOR
2948:S2CID
2917:S2CID
2891:arXiv
2857:S2CID
2818:JSTOR
2787:JSTOR
2762:JSTOR
2717:(PDF)
2690:(PDF)
2542:JSTOR
2508:(PDF)
2497:(PDF)
2479:S2CID
2463:JSTOR
2056:Notes
2029:Plato
1383:Euler
1367:limit
383:most
111:than
5579:Evil
5259:Self
5209:and
5099:Blum
4810:and
4614:Lobb
4569:Cake
4564:Bell
4314:Star
4221:Ulam
4120:Pell
3909:Cube
3726:Base
3318:PMID
3240:= ±1
3218:The
3034:CACM
2668:ISBN
2640:ISBN
2567:ISBN
2195:See
1767:lazy
1559:1981
1555:1976
1391:grid
1304:and
402:OEIS
5697:Ban
5085:or
4604:Lah
3368:doi
3338:doi
3310:doi
3273:doi
3263:",
3250:(2)
3190:doi
3143:doi
3062:doi
3016:doi
2976:doi
2966:",
2940:doi
2901:doi
2887:113
2849:doi
2810:doi
2754:doi
2729:doi
2608:doi
2534:doi
2455:doi
2329:or
2031:'s
1906:.
1845:to
1522:).
1385:'s
1336:In
931:log
925:log
887:log
795:is
738:log
721:log
704:log
319:max
160:In
138:In
127:or
89:In
47:of
5767::
3374:,
3364:27
3362:,
3344:,
3334:22
3332:,
3316:,
3308:,
3298:18
3296:,
3279:,
3269:13
3267:,
3246:,
3238:Dy
3216:,
3196:,
3186:78
3184:,
3165:,
3159:MR
3157:,
3149:,
3137:,
3120:MR
3096:18
3094:,
3058:20
3056:,
3037:19
3022:,
3012:15
3010:,
3004:,
2982:,
2972:41
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