6073:
1258:. A rhyme scheme describes which lines rhyme with each other, and so may be interpreted as a partition of the set of lines into rhyming subsets. Rhyme schemes are usually written as a sequence of Roman letters, one per line, with rhyming lines given the same letter as each other, and with the first lines in each rhyming set labeled in alphabetical order. Thus, the 15 possible four-line rhyme schemes are AAAA, AAAB, AABA, AABB, AABC, ABAA, ABAB, ABAC, ABBA, ABBB, ABBC, ABCA, ABCB, ABCC, and ABCD.
440:
456:
5997:
9575:
5525:
1328:
1-32, 3-12, 21-3, and 23-1 are also counted by the Bell numbers. The permutations in which every 321 pattern (without restriction on consecutive values) can be extended to a 3241 pattern are also counted by the Bell numbers. However, the Bell numbers grow too quickly to count the permutations that avoid a pattern that has not been generalized in this way: by the (now proven)
5715:
3432:, a style of mathematical reasoning in which sets of mathematical objects are described by formulas explaining their construction from simpler objects, and then those formulas are manipulated to derive the combinatorial properties of the objects. In the language of analytic combinatorics, a set partition may be described as a set of nonempty
5109:
1327:
where values that must be consecutive are written adjacent to each other, and values that can appear non-consecutively are separated by a dash, these permutations can be described as the permutations that avoid the pattern 1-23. The permutations that avoid the generalized patterns 12-3, 32-1, 3-21,
2835:
5992:{\displaystyle {\begin{aligned}{\frac {\ln B_{n}}{n}}&=\ln n-\ln \ln n-1+{\frac {\ln \ln n}{\ln n}}+{\frac {1}{\ln n}}+{\frac {1}{2}}\left({\frac {\ln \ln n}{\ln n}}\right)^{2}+O\left({\frac {\ln \ln n}{(\ln n)^{2}}}\right)\\&{}\qquad {\text{as }}n\to \infty \end{aligned}}}
5095:
1782:
3061:
3424:
In this formula, the summation in the middle is the general form used to define the exponential generating function for any sequence of numbers, and the formula on the right is the result of performing the summation in the specific case of the Bell numbers.
6094:. Bell did not claim to have discovered these numbers; in his 1938 paper, he wrote that the Bell numbers "have been frequently investigated" and "have been rediscovered many times". Bell cites several earlier publications on these numbers, beginning with
2259:
5520:{\displaystyle B_{n+h}={\frac {(n+h)!}{W(n)^{n+h}}}\times {\frac {\exp(e^{W(n)}-1)}{(2\pi B)^{1/2}}}\times \left(1+{\frac {P_{0}+hP_{1}+h^{2}P_{2}}{e^{W(n)}}}+{\frac {Q_{0}+hQ_{1}+h^{2}Q_{2}+h^{3}Q_{3}+h^{4}Q_{4}}{e^{2W(n)}}}+O(e^{-3W(n)})\right)}
2561:
2100:
nonempty subsets. Thus, in the equation relating the Bell numbers to the
Stirling numbers, each partition counted on the left hand side of the equation is counted in exactly one of the terms of the sum on the right hand side, the one for which
3524:
3286:
2574:
4404:
991:. The equivalence relation corresponding to a partition defines two elements as being equivalent when they belong to the same partition subset as each other. Conversely, every equivalence relation corresponds to a partition into
4822:
4726:
3419:
3817:
4945:
6126:
sprang up, in which guests were given five packets of incense to smell and were asked to guess which ones were the same as each other and which were different. The 52 possible solutions, counted by the Bell number
2385:
4535:
3656:
An alternative method for deriving the same generating function uses the recurrence relation for the Bell numbers in terms of binomial coefficients to show that the exponential generating function satisfies the
1232:
1489:
Determine the numbers not on the left column by taking the sum of the number to the left and the number above the number to the left, that is, the number diagonally up and left of the number we are calculating
4957:
2052:
963:
that all of the subsets in this family are non-empty subsets of the empty set and that they are pairwise disjoint subsets of the empty set, because there are no subsets to have these unlikely properties.
4085:
2870:
1900:
1130:
into numbers greater than one, treating two factorizations as the same if they have the same factors in a different order. For instance, 30 is the product of the three primes 2, 3, and 5, and has
1583:
3985:
3591:
1681:
7155:
5720:
236:
2394:
3651:
3621:
1476:
2090:
4627:
3122:
1670:
4157:
705:
5601:
3717:
867:
813:
759:
7677:
4585:
2117:
1960:
3551:
915:
4224:
159:
5575:
4186:
3454:
1415:
643:
605:
1278:
cards is shuffled by repeatedly removing the top card and reinserting it anywhere in the deck (including its original position at the top of the deck), with exactly
5675:
5648:
1617:
1155:
1080:
953:
488:
390:
328:
277:
85:
5704:
4244:
3115:
3089:
2863:
4296:
5621:
4555:
1128:
1104:
1049:
1021:
572:
552:
532:
512:
410:
352:
297:
105:
2273:
4734:
4635:
6082:
are based on the 52 ways of partitioning five elements (the two red symbols represent the same partition, and the green symbol is added for reaching 54).
3623:
describes the overall partition as a set of these urns. The exponential generating function may then be read off from this notation by translating the
6612:
6296:
6050:
6036:
4274:
4253:
245:
6497:
3310:
2830:{\displaystyle \sum _{j=0}^{n}{\binom {n}{j}}a^{j}b^{n-j}B_{j}=\sum _{i=0}^{k}\left(-1)^{k-i}\sum _{j=0}^{n}{\binom {n}{j}}a^{j}(b-ak)^{n-j}B_{j+i},}
7670:
3736:
4102: = 2, the Bell numbers repeat the pattern odd-odd-even with period three. The period of this repetition, for an arbitrary prime number
4827:
7211:
6116:
The first exhaustive enumeration of set partitions appears to have occurred in medieval Japan, where (inspired by the popularity of the book
1347:
959:. This partition is itself the empty set; it can be interpreted as a family of subsets of the empty set, consisting of zero subsets. It is
4460:
1163:
5090:{\displaystyle B_{n}\sim {\frac {1}{\sqrt {n}}}\left({\frac {n}{W(n)}}\right)^{n+{\frac {1}{2}}}\exp \left({\frac {n}{W(n)}}-n-1\right).}
8477:
7663:
1977:
49:. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of
8472:
6177:
2566:
1985:
8487:
8467:
7547:
7327:
7101:
6907:
6222:
1323:
items in which no three values that are in sorted order have the last two of these three consecutive. In a notation for generalized
3996:
1813:
9180:
8760:
4422:
7374:
7178:
6134:, were recorded by 52 different diagrams, which were printed above the chapter headings in some editions of The Tale of Genji.
1493:
6029:
2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837 (sequence
1332:, the number of such permutations is singly exponential, and the Bell numbers have a higher asymptotic growth rate than that.
8482:
1777:{\displaystyle {\begin{array}{l}1\\1&2\\2&3&5\\5&7&10&15\\15&20&27&37&52\end{array}}}
9266:
3909:
1286:
different shuffles that can be performed. Of these, the number that return the deck to its original sorted order is exactly
7571:
7409:
7278:
3556:
2265:
8582:
7116:
8932:
8251:
8044:
6794:
6770:
4287:
8967:
8937:
8612:
8602:
1329:
167:
9108:
8522:
8256:
8236:
7273:
6072:
8798:
3553:
is a combinatorial class with only a single member of size one, an element that can be placed into an urn. The inner
3056:{\displaystyle \sum _{j=0}^{n}{\binom {n}{j}}a^{j}b^{n-j}B_{j}=\sum _{j=0}^{n}{\binom {n}{j}}a^{j}(b-a)^{n-j}B_{j+1}}
50:
8962:
9599:
9057:
8680:
8437:
8246:
8228:
8122:
8112:
8102:
7491:
4410:
8942:
9185:
8730:
8351:
8137:
8132:
8127:
8117:
8094:
6950:
3626:
3596:
1423:
6102:
for the Bell numbers. Bell called these numbers "exponential numbers"; the name "Bell numbers" and the notation
2060:
8170:
3892:
3873:
1083:
362:
8427:
6099:
4590:
1629:
9296:
9261:
9047:
8957:
8831:
8806:
8715:
8705:
8317:
8299:
8219:
6917:
Berend, D.; Tassa, T. (2010). "Improved bounds on Bell numbers and on moments of sums of random variables".
4112:
651:
9556:
8826:
8700:
8331:
8107:
7887:
7814:
7369:
7215:
6506:
5580:
3429:
2254:{\displaystyle B_{n+m}=\sum _{k=0}^{n}\sum _{j=0}^{m}\left\{{m \atop j}\right\}{n \choose k}j^{n-k}B_{k}.}
1371:
7313:
8811:
8665:
8592:
7747:
3658:
2556:{\displaystyle \sum _{j=0}^{n}{\binom {n}{j}}B_{k+j}=\sum _{i=0}^{k}{\binom {k}{i}}(-1)^{k-i}B_{n+i+1}.}
1909: + 1 items, removing the set containing the first item leaves a partition of a smaller set of
819:
765:
711:
9520:
9160:
6714:
Asai, Nobuhiro; Kubo, Izumi; Kuo, Hui-Hsiung (2000). "Bell numbers, log-concavity, and log-convexity".
4564:
3728:
1587:
Repeat step three until there is a new row with one more number than the previous row (do step 3 until
1924:
439:
9453:
9347:
9311:
9052:
8775:
8755:
8572:
8241:
8029:
8001:
7500:
7462:
7286:
Hurst, Greg; Schultz, Andrew (2009). "An elementary (number theory) proof of
Touchard's congruence".
7243:
6985:
6162:
6141:'s second notebook, he investigated both Bell polynomials and Bell numbers. Early references for the
3869:
3842:
3838:
3663:
1804:
972:
920:
As suggested by the set notation above, the ordering of subsets within the family is not considered;
413:
358:
300:
7645:
7447:
3532:
873:
443:
Partitions of sets can be arranged in a partial order, showing that each partition of a set of size
9175:
9039:
9034:
9002:
8765:
8740:
8735:
8710:
8640:
8636:
8567:
8457:
8289:
8085:
8054:
7234:
7232:(1978). "The Bells: versatile numbers that can count partitions of a set, primes and even rhymes".
7221:
6138:
4447:
4191:
3653:
operator into the exponential function and the nonemptiness constraint ≥1 into subtraction by one.
3441:
3301:
1800:
1324:
988:
925:
118:
9574:
7268:
3519:{\displaystyle \mathrm {S\scriptstyle ET} (\mathrm {S\scriptstyle ET} _{\geq 1}({\mathcal {Z}})).}
9578:
9332:
9327:
9241:
9215:
9113:
9092:
8864:
8745:
8695:
8617:
8587:
8527:
8294:
8274:
8205:
7918:
7588:
7426:
7391:
7287:
7112:
7079:
7049:
6975:
6880:
6847:
6814:
6749:
6723:
5533:
4948:
1293:. Thus, the probability that the deck is in its original order after shuffling it in this way is
984:
980:
491:
434:
46:
8462:
7176:
Engel, Konrad (1994). "On the average rank of an element in a filter of the partition lattice".
3281:{\displaystyle \sum _{j=0}^{n}{\binom {n}{j}}B_{j}k^{n-j}=\sum _{i=0}^{k}\leftB_{n+i}(-1)^{k-i}}
6339:
4165:
1381:
610:
9472:
9417:
9271:
9246:
9220:
8997:
8675:
8670:
8597:
8577:
8562:
8284:
8266:
8185:
8175:
8160:
7938:
7923:
7626:
7543:
7323:
7207:
7097:
6903:
6361:
6218:
6118:
992:
7317:
6212:
577:
357:
As well as appearing in counting problems, these numbers have a different interpretation, as
9508:
9301:
8887:
8859:
8849:
8841:
8725:
8690:
8685:
8652:
8346:
8309:
8200:
8195:
8190:
8180:
8152:
8039:
7991:
7986:
7943:
7882:
7580:
7553:
7508:
7418:
7383:
7333:
7251:
7187:
7093:
7087:
7059:
7018:
6931:
6872:
6860:
6839:
6827:
6806:
6779:
6761:
6733:
6351:
6217:. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg. pp. 27–28.
6091:
6087:
1367:
54:
7600:
7522:
7474:
7438:
7361:
7199:
7071:
7032:
6997:
6745:
6232:
5653:
5626:
4399:{\displaystyle B_{n}={\frac {n!}{2\pi ie}}\int _{\gamma }{\frac {e^{e^{z}}}{z^{n+1}}}\,dz.}
3593:
operator describes a set or urn that contains one or more labelled elements, and the outer
1590:
1133:
1058:
931:
466:
455:
368:
306:
255:
63:
9484:
9373:
9306:
9232:
9155:
9129:
8947:
8660:
8517:
8452:
8422:
8412:
8407:
8073:
7981:
7928:
7772:
7712:
7596:
7557:
7518:
7470:
7434:
7404:
7357:
7337:
7304:(2013). "Two thousand years of combinatorics". In Wilson, Robin; Watkins, John J. (eds.).
7195:
7067:
7028:
6993:
6741:
6228:
6172:
5680:
3861:
3826:
for the exponential function, and then collecting terms with the same exponent. It allows
3289:
976:
7262:
Fractal Music, Hypercards, and more ... Mathematical
Recreations from Scientific American
4229:
3094:
3068:
2842:
1420:
Start a new row with the rightmost element from the previous row as the leftmost number (
7504:
7466:
7247:
6989:
9489:
9357:
9342:
9206:
9170:
9145:
9021:
8992:
8977:
8854:
8750:
8720:
8447:
8402:
8279:
7877:
7872:
7867:
7839:
7824:
7737:
7722:
7700:
7687:
7229:
7083:
6894:
6688:
also mention the connection between Bell numbers and The Tale of Genji, in less detail.
6167:
5606:
4540:
3846:
3822:
This formula can be derived by expanding the exponential generating function using the
1113:
1089:
1034:
1006:
557:
537:
517:
497:
395:
337:
282:
90:
7255:
9593:
9412:
9396:
9337:
9291:
8987:
8972:
8882:
8607:
8165:
8034:
7996:
7953:
7834:
7819:
7809:
7767:
7757:
7732:
7191:
7023:
7006:
6579:
6142:
6078:
4409:
Some asymptotic representations can then be derived by a standard application of the
3823:
1355:
1341:
1107:
960:
921:
38:
6753:
4817:{\displaystyle ~n_{0}(\varepsilon )=\max \left\{e^{4},d^{-1}(\varepsilon )\right\}~}
4721:{\displaystyle B_{n}<\left({\frac {e^{-0.6+\varepsilon }n}{\ln(n+1)}}\right)^{n}}
9448:
9437:
9352:
9190:
9165:
9082:
8982:
8952:
8927:
8911:
8816:
8783:
8532:
8506:
8417:
8356:
7933:
7829:
7762:
7742:
7717:
7530:
7486:
7482:
7301:
6018:
3900:
1243:
1052:
331:
7617:
7513:
6090:, who wrote about them in 1938, following up a 1934 paper in which he studied the
4290:
to the exponential generating function yields the complex integral representation
1346:
1311:
Related to card shuffling are several other problems of counting special kinds of
9407:
9282:
9087:
8551:
8442:
8397:
8142:
8049:
7948:
7777:
7752:
7727:
7629:
7534:
7487:"Aurifeuillian factorizations and the period of the Bell numbers modulo a prime"
6602:
6286:
3433:
1350:
The triangular array whose right-hand diagonal sequence consists of Bell numbers
1312:
31:
6893:
Bender, Edward A.; Williamson, S. Gill (2006). "Example 11.7, Set
Partitions".
3414:{\displaystyle B(x)=\sum _{n=0}^{\infty }{\frac {B_{n}}{n!}}x^{n}=e^{e^{x}-1}.}
9544:
9525:
8821:
8432:
6784:
6765:
6737:
1308:! probability that would describe a uniformly random permutation of the deck.
1024:
7655:
6365:
4267:
1, 3, 13, 12, 781, 39, 137257, 24, 39, 2343, 28531167061, 156, ... (sequence
3812:{\displaystyle B_{n}={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {k^{n}}{k!}}.}
9150:
9077:
9069:
8874:
8788:
7906:
7634:
7345:
6932:"Ramanujan Reaches His Hand From His Grave To Snatch Your Theorems From You"
1267:
968:
956:
17:
7063:
4940:{\displaystyle ~d(x):=\ln \ln(x+1)-\ln \ln x+{\frac {1+e^{-1}}{\ln x}}\,.}
1786:
The Bell numbers appear on both the left and right sides of the triangle.
9251:
7086:(1996). "Famous Families of Numbers: Bell Numbers and Stirling Numbers".
3880:
1686:
1676:
Here are the first five rows of the triangle constructed by these rules:
6356:
9256:
8915:
7592:
7430:
7395:
6884:
6851:
6818:
4090:
Because of
Touchard's congruence, the Bell numbers are periodic modulo
1976:
A different summation formula represents each Bell number as a sum of
1028:
108:
7260:
Reprinted with an addendum as "The Tinkly Temple Bells", Chapter 2 of
2380:{\displaystyle B_{n}=\sum _{k=0}^{n}{\binom {n}{k}}(-1)^{n-k}B_{k+1},}
1905:
It can be explained by observing that, from an arbitrary partition of
7054:
6980:
6964:"A combinatorial interpretation of the eigensequence for composition"
6728:
6496:
Simon, Barry (2010). "Example 15.4.6 (Asymptotics of Bell
Numbers)".
6017:
raised the question of whether infinitely many Bell numbers are also
1255:
7584:
7422:
7387:
6876:
6843:
6810:
4530:{\displaystyle B_{n}<\left({\frac {0.792n}{\ln(n+1)}}\right)^{n}}
1354:
The Bell numbers can easily be calculated by creating the so-called
1227:{\displaystyle 30=2\times 15=3\times 10=5\times 6=2\times 3\times 5}
995:. Therefore, the Bell numbers also count the equivalence relations.
7348:; Wyman, Max (1955). "An asymptotic formula for the Bell numbers".
7322:(2nd ed.). Amsterdam, Netherlands: North-Holland. p. 17.
7292:
6071:
1345:
454:
438:
534:
is defined as a family of nonempty, pairwise disjoint subsets of
6377:
6375:
2047:{\displaystyle B_{n}=\sum _{k=0}^{n}\left\{{n \atop k}\right\}.}
1251:
417:
112:
9542:
9506:
9470:
9434:
9394:
9019:
8908:
8634:
8549:
8504:
8381:
8071:
8018:
7970:
7904:
7856:
7794:
7698:
7659:
6419:
6417:
6963:
6043:
corresponding to the indices 2, 3, 7, 13, 42 and 55 (sequence
1315:
that are also answered by the Bell numbers. For instance, the
279:
counts the different ways to partition a set that has exactly
4951:, a function with the same growth rate as the logarithm, as
3719:. The function itself can be found by solving this equation.
6076:
The traditional
Japanese symbols for the 54 chapters of the
4080:{\displaystyle B_{p^{m}+n}\equiv mB_{n}+B_{n+1}{\pmod {p}}.}
3538:
3502:
2111:
has given a formula that combines both of these summations:
1895:{\displaystyle B_{n+1}=\sum _{k=0}^{n}{\binom {n}{k}}B_{k}.}
7565:
Williams, G. T. (1945). "Numbers generated by the function
6606:
6290:
6145:, which has the Bell numbers on both of its sides, include
6045:
6031:
4269:
4248:
240:
7040:
Claesson, Anders (2001). "Generalized pattern avoidance".
3856:
th Bell number is also the sum of the coefficients in the
1378:
Start with the number one. Put this on a row by itself. (
7646:"Further properties & Generalization of Bell-Numbers"
1578:{\displaystyle (x_{i,j}\leftarrow x_{i,j-1}+x_{i-1,j-1})}
7350:
Transactions of the Royal
Society of Canada, Section III
2092:
is the number of ways to partition a set of cardinality
7214:(2009). "II.3 Surjections, set partitions, and words".
7161: = 1, 2, 3, 4, 5, …"
6797:(1948). "The arithmetic of Bell and Stirling numbers".
1622:
The number on the left hand side of a given row is the
7120:
3980:{\displaystyle B_{p+n}\equiv B_{n}+B_{n+1}{\pmod {p}}}
3634:
3604:
3565:
3479:
3462:
1929:
7119:
6194:
6192:
5718:
5683:
5656:
5629:
5609:
5583:
5536:
5112:
4960:
4830:
4737:
4638:
4593:
4567:
4543:
4463:
4299:
4232:
4194:
4168:
4115:
3999:
3912:
3739:
3666:
3629:
3599:
3559:
3535:
3457:
3313:
3125:
3097:
3071:
2873:
2845:
2577:
2397:
2276:
2120:
2063:
1988:
1927:
1816:
1684:
1632:
1593:
1496:
1426:
1384:
1166:
1136:
1116:
1092:
1061:
1037:
1009:
934:
876:
822:
768:
714:
654:
613:
580:
560:
540:
520:
500:
469:
398:
371:
340:
309:
285:
258:
170:
121:
93:
66:
6625:
6623:
4947:
The Bell numbers can also be approximated using the
3586:{\displaystyle \mathrm {S\scriptstyle ET} _{\geq 1}}
1319:
th Bell number equals the number of permutations on
924:
are counted by a different sequence of numbers, the
9366:
9320:
9280:
9231:
9205:
9138:
9122:
9101:
9068:
9033:
8873:
8840:
8797:
8774:
8651:
8339:
8330:
8308:
8265:
8227:
8218:
8151:
8093:
8084:
7150:{\displaystyle \textstyle \sum {\frac {n^{m}}{n!}}}
6668:. However, Rota gives an incorrect date, 1934, for
6392:
6390:
955:is 1 because there is exactly one partition of the
447:"uses" one of the partitions of a set of size
7149:
5991:
5698:
5669:
5642:
5615:
5595:
5569:
5519:
5089:
4939:
4816:
4720:
4621:
4579:
4549:
4529:
4398:
4238:
4218:
4180:
4151:
4079:
3979:
3811:
3711:
3645:
3615:
3585:
3545:
3518:
3413:
3280:
3109:
3083:
3055:
2857:
2829:
2555:
2379:
2253:
2084:
2046:
1954:
1894:
1776:
1664:
1611:
1577:
1470:
1409:
1226:
1149:
1122:
1098:
1074:
1043:
1015:
947:
909:
861:
807:
753:
699:
637:
599:
566:
546:
526:
506:
482:
404:
384:
346:
322:
291:
271:
230:
153:
99:
79:
6381:
3163:
3150:
2993:
2980:
2911:
2898:
2761:
2748:
2615:
2602:
2497:
2484:
2435:
2422:
2327:
2314:
2216:
2203:
1873:
1860:
231:{\displaystyle 1,1,2,5,15,52,203,877,4140,\dots }
6423:
4763:
1966:items that remain after one set is removed, and
1031:, meaning that it is the product of some number
6580:"The Moser-Wyman expansion of the Bell numbers"
6064:, which is approximately 9.30740105 × 10.
6896:Foundations of Combinatorics with Applications
6248:credits this observation to Silvio Minetola's
1282:repetitions of this operation, then there are
7671:
7407:(1964). "The number of partitions of a set".
6863:(1938). "The iterated exponential integers".
6669:
6447:
6291:"Sequence A011971 (Aitken's array)"
6110:
4450:formulas for the Bell numbers are known. In
4438:!, gives a logarithmically concave sequence.
1945:
1932:
8:
7542:(2nd ed.). Boston, MA: Academic Press.
6459:
901:
898:
880:
877:
853:
850:
838:
832:
826:
823:
799:
796:
784:
778:
772:
769:
745:
742:
730:
724:
718:
715:
691:
688:
682:
676:
670:
664:
658:
655:
632:
614:
7448:"A generalized recurrence for Bell numbers"
6553:
6338:Komatsu, Takao; Pita-Ruiz, Claudio (2018).
4451:
3646:{\displaystyle \mathrm {S\scriptstyle ET} }
3616:{\displaystyle \mathrm {S\scriptstyle ET} }
1471:{\displaystyle x_{i,1}\leftarrow x_{i-1,r}}
1304:, which is significantly larger than the 1/
9539:
9503:
9467:
9431:
9391:
9065:
9030:
9016:
8905:
8648:
8631:
8546:
8501:
8378:
8336:
8224:
8090:
8081:
8068:
8015:
7972:Possessing a specific set of other numbers
7967:
7901:
7853:
7791:
7695:
7678:
7664:
7656:
5100:
2085:{\displaystyle \left\{{n \atop k}\right\}}
459:The 52 partitions of a set with 5 elements
7512:
7308:. Oxford University Press. pp. 7–37.
7291:
7130:
7124:
7118:
7053:
7022:
6979:
6783:
6727:
6613:On-Line Encyclopedia of Integer Sequences
6355:
6322:
6297:On-Line Encyclopedia of Integer Sequences
6003:
5971:
5968:
5950:
5914:
5898:
5862:
5847:
5826:
5791:
5736:
5723:
5719:
5717:
5682:
5661:
5655:
5634:
5628:
5608:
5582:
5535:
5488:
5455:
5444:
5434:
5421:
5411:
5398:
5388:
5375:
5359:
5352:
5332:
5321:
5311:
5298:
5282:
5275:
5248:
5244:
5202:
5186:
5168:
5132:
5117:
5111:
5046:
5023:
5016:
4991:
4974:
4965:
4959:
4933:
4910:
4897:
4829:
4788:
4775:
4745:
4736:
4712:
4664:
4657:
4643:
4637:
4604:
4592:
4566:
4542:
4521:
4482:
4468:
4462:
4386:
4372:
4360:
4355:
4349:
4343:
4313:
4304:
4298:
4259:The period of the Bell numbers to modulo
4231:
4193:
4167:
4123:
4116:
4114:
4058:
4046:
4033:
4009:
4004:
3998:
3961:
3949:
3936:
3917:
3911:
3790:
3784:
3778:
3767:
3753:
3744:
3738:
3691:
3665:
3630:
3628:
3600:
3598:
3574:
3561:
3558:
3537:
3536:
3534:
3501:
3500:
3488:
3475:
3458:
3456:
3394:
3389:
3376:
3356:
3350:
3344:
3333:
3312:
3266:
3241:
3223:
3213:
3202:
3183:
3173:
3162:
3149:
3147:
3141:
3130:
3124:
3096:
3070:
3041:
3025:
3003:
2992:
2979:
2977:
2971:
2960:
2947:
2931:
2921:
2910:
2897:
2895:
2889:
2878:
2872:
2844:
2812:
2796:
2771:
2760:
2747:
2745:
2739:
2728:
2712:
2685:
2675:
2664:
2651:
2635:
2625:
2614:
2601:
2599:
2593:
2582:
2576:
2532:
2516:
2496:
2483:
2481:
2475:
2464:
2445:
2434:
2421:
2419:
2413:
2402:
2396:
2389:which can be generalized in this manner:
2362:
2346:
2326:
2313:
2311:
2305:
2294:
2281:
2275:
2242:
2226:
2215:
2202:
2200:
2186:
2176:
2165:
2155:
2144:
2125:
2119:
2068:
2062:
2027:
2017:
2006:
1993:
1987:
1944:
1931:
1928:
1926:
1883:
1872:
1859:
1857:
1851:
1840:
1821:
1815:
1685:
1683:
1650:
1637:
1631:
1592:
1548:
1523:
1504:
1495:
1450:
1431:
1425:
1389:
1383:
1165:
1141:
1135:
1115:
1091:
1066:
1060:
1036:
1008:
939:
933:
875:
821:
767:
713:
653:
612:
585:
579:
559:
539:
519:
499:
474:
468:
397:
376:
370:
339:
314:
308:
284:
263:
257:
169:
139:
126:
120:
92:
71:
65:
27:Count of the possible partitions of a set
7092:. Copernicus Series. Springer. pp.
6541:
6483:
6471:
6435:
6261:
6245:
6095:
4622:{\displaystyle n>n_{0}(\varepsilon )}
2105:is the number of sets in the partition.
7220:. Cambridge University Press. pp.
6919:Probability and Mathematical Statistics
6681:
6198:
6188:
6109:for these numbers was given to them by
6014:
4454:the following bounds were established:
3436:into which elements labelled from 1 to
1665:{\displaystyle B_{i}\leftarrow x_{i,1}}
1271:
645:can be partitioned in 5 distinct ways:
6697:
6685:
6565:
6273:
6150:
6146:
4152:{\displaystyle {\frac {p^{p}-1}{p-1}}}
3290:inversion formula for Stirling numbers
2268:to the recurrence relation, we obtain
2108:
700:{\displaystyle \{\{a\},\{b\},\{c\}\},}
7264:, W. H. Freeman, 1992, pp. 24–38
7007:"Engel's inequality for Bell numbers"
6629:
6529:
1270:problem mentioned in the addendum to
57:, who wrote about them in the 1930s.
7:
7319:Combinatorial Problems and Exercises
6665:
6653:
6641:
6408:
6396:
6333:
6331:
6310:
5596:{\displaystyle n\rightarrow \infty }
4246:it is exactly this number (sequence
3723:Moments of probability distributions
7372:(1880). "On the algebra of logic".
7316:(1993). "Section 1.14, Problem 9".
6957:(3rd ed.). Dover. p. 108.
6939:Asia Pacific Mathematics Newsletter
6830:(1934). "Exponential polynomials".
4425:. Dividing them by the factorials,
4066:
3969:
3448:) may be expressed by the notation
3428:One way to derive this result uses
1978:Stirling numbers of the second kind
1266:The Bell numbers come up in a card
6505:. pp. 772–774. Archived from
6178:Stirling numbers of the first kind
5982:
5590:
3779:
3638:
3635:
3631:
3608:
3605:
3601:
3569:
3566:
3562:
3483:
3480:
3476:
3466:
3463:
3459:
3345:
3224:
3154:
2984:
2902:
2752:
2686:
2606:
2567:Stirling numbers of the first kind
2488:
2426:
2318:
2207:
2187:
2069:
2028:
1973:choices of how to partition them.
1936:
1864:
862:{\displaystyle \{\{c\},\{a,b\}\},}
808:{\displaystyle \{\{b\},\{a,c\}\},}
754:{\displaystyle \{\{a\},\{b,c\}\},}
25:
7306:Combinatorics: Ancient and Modern
7256:10.1038/scientificamerican0578-24
7042:European Journal of Combinatorics
6250:Principii di Analisi Combinatoria
6086:The Bell numbers are named after
6025:. The first few Bell primes are:
4580:{\displaystyle \varepsilon >0}
161:, the first few Bell numbers are
9573:
9181:Perfect digit-to-digit invariant
6340:"Some formulas for Bell numbers"
2565:Other finite sum formulas using
1955:{\displaystyle {\tbinom {n}{k}}}
1336:Triangle scheme for calculations
1242:The Bell numbers also count the
7375:American Journal of Mathematics
7179:Journal of Combinatorial Theory
7011:Journal of Combinatorial Theory
6799:American Journal of Mathematics
5970:
4423:logarithmically convex sequence
4059:
3962:
3712:{\displaystyle B'(x)=e^{x}B(x)}
3440:have been distributed, and the
3302:exponential generating function
299:elements, or equivalently, the
6955:Asymptotic methods in analysis
5979:
5947:
5934:
5693:
5687:
5587:
5564:
5561:
5555:
5546:
5509:
5504:
5498:
5481:
5468:
5462:
5342:
5336:
5241:
5228:
5223:
5212:
5206:
5195:
5165:
5158:
5147:
5135:
5061:
5055:
5006:
5000:
4873:
4861:
4843:
4837:
4803:
4797:
4757:
4751:
4702:
4690:
4616:
4610:
4511:
4499:
4070:
4060:
3973:
3963:
3706:
3700:
3681:
3675:
3546:{\displaystyle {\mathcal {Z}}}
3510:
3507:
3497:
3471:
3323:
3317:
3263:
3253:
3022:
3009:
2793:
2777:
2709:
2699:
2513:
2503:
2343:
2333:
1643:
1572:
1516:
1497:
1443:
1082:gives the number of different
910:{\displaystyle \{\{a,b,c\}\}.}
1:
8020:Expressible via specific sums
7572:American Mathematical Monthly
7514:10.1090/S0025-5718-96-00683-7
7410:American Mathematical Monthly
6716:Acta Applicandae Mathematicae
6382:Flajolet & Sedgewick 2009
4219:{\displaystyle p=113,163,167}
3292:applied to Spivey’s formula.
154:{\displaystyle B_{0}=B_{1}=1}
60:The Bell numbers are denoted
7455:Journal of Integer Sequences
7192:10.1016/0097-3165(94)90038-8
7024:10.1016/0097-3165(95)90033-0
7005:Canfield, E. Rodney (1995).
6968:Journal of Integer Sequences
6424:Bender & Williamson 2006
9109:Multiplicative digital root
7446:Spivey, Michael Z. (2008).
7274:Encyclopedia of Mathematics
6902:. Dover. pp. 319–320.
6766:"A problem in combinations"
6578:Canfield, Rod (July 1994).
6448:Becker & Riordan (1948)
5570:{\displaystyle h=O(\ln(n))}
3876:as a function of the first
3444:of all partitions (for all
2839:which simplifies down with
1799:The Bell numbers satisfy a
9616:
7618:"Diagrams of Bell numbers"
7492:Mathematics of Computation
6603:Sloane, N. J. A.
6460:Hurst & Schultz (2009)
6287:Sloane, N. J. A.
5709:The asymptotic expression
5103:established the expansion
4537:for all positive integers
4411:method of steepest descent
2266:Pascal's inversion formula
1339:
607:because the 3-element set
432:
330:also counts the different
29:
9569:
9552:
9538:
9516:
9502:
9480:
9466:
9444:
9430:
9403:
9390:
9186:Perfect digital invariant
9029:
9015:
8923:
8904:
8761:Superior highly composite
8647:
8630:
8558:
8545:
8513:
8500:
8388:
8377:
8080:
8067:
8025:
8014:
7977:
7966:
7914:
7900:
7863:
7852:
7805:
7790:
7708:
7694:
6930:Berndt, Bruce C. (2011).
6785:10.1017/S1757748900002334
6670:Becker & Riordan 1948
6554:Asai, Kubo & Kuo 2000
6111:Becker & Riordan 1948
5677:are known expressions in
4288:Cauchy's integral formula
4181:{\displaystyle p\leq 101}
4094:, for every prime number
3833:to be interpreted as the
3727:The Bell numbers satisfy
3288:which can be seen as the
1917:that may range from 0 to
1410:{\displaystyle x_{0,1}=1}
1084:multiplicative partitions
638:{\displaystyle \{a,b,c\}}
363:probability distributions
111:greater than or equal to
39:combinatorial mathematics
8799:Euler's totient function
8583:Euler–Jacobi pseudoprime
7858:Other polynomial numbers
7461:(2): Article 08.2.5, 3.
6211:Halmos, Paul R. (1974).
4421:The Bell numbers form a
3874:probability distribution
3862:complete Bell polynomial
1482:is the last element of (
967:The partitions of a set
51:Stigler's law of eponymy
30:Not to be confused with
8613:Somer–Lucas pseudoprime
8603:Lucas–Carmichael number
8438:Lazy caterer's sequence
7536:Generatingfunctionology
6738:10.1023/A:1010738827855
6607:"Sequence A051131"
6323:Conway & Guy (1996)
6122:) a parlor game called
4452:Berend & Tassa 2010
4282:Integral representation
4106:, must be a divisor of
3304:of the Bell numbers is
1330:Stanley–Wilf conjecture
600:{\displaystyle B_{3}=5}
514:. A partition of a set
53:, they are named after
8488:Wedderburn–Etherington
7888:Lucky numbers of Euler
7217:Analytic Combinatorics
7151:
7064:10.1006/eujc.2001.0515
6962:Callan, David (2006).
6083:
5993:
5700:
5671:
5644:
5617:
5597:
5571:
5521:
5101:Moser & Wyman 1955
5091:
4941:
4818:
4722:
4623:
4581:
4551:
4531:
4400:
4240:
4220:
4182:
4153:
4081:
3981:
3891:The Bell numbers obey
3864:, which expresses the
3813:
3783:
3713:
3647:
3617:
3587:
3547:
3520:
3430:analytic combinatorics
3415:
3349:
3282:
3218:
3146:
3111:
3085:
3057:
2976:
2894:
2859:
2831:
2744:
2680:
2598:
2557:
2480:
2418:
2381:
2310:
2255:
2181:
2160:
2086:
2048:
2022:
1956:
1913:items for some number
1896:
1856:
1778:
1666:
1613:
1579:
1472:
1411:
1372:Charles Sanders Peirce
1351:
1228:
1151:
1124:
1100:
1076:
1045:
1017:
949:
911:
863:
809:
755:
701:
639:
601:
568:
548:
528:
508:
484:
460:
452:
406:
386:
348:
324:
293:
273:
232:
155:
101:
81:
8776:Prime omega functions
8593:Frobenius pseudoprime
8383:Combinatorial numbers
8252:Centered dodecahedral
8045:Primary pseudoperfect
7152:
7113:"Summirung der Reihe
7111:Dobiński, G. (1877).
6865:Annals of Mathematics
6832:Annals of Mathematics
6075:
5994:
5701:
5672:
5670:{\displaystyle Q_{i}}
5645:
5643:{\displaystyle P_{i}}
5618:
5598:
5572:
5522:
5092:
4942:
4819:
4723:
4624:
4582:
4552:
4532:
4401:
4241:
4221:
4183:
4154:
4082:
3982:
3893:Touchard's congruence
3814:
3763:
3714:
3659:differential equation
3648:
3618:
3588:
3548:
3521:
3416:
3329:
3283:
3198:
3126:
3112:
3086:
3058:
2956:
2874:
2860:
2832:
2724:
2660:
2578:
2558:
2460:
2398:
2382:
2290:
2256:
2161:
2140:
2087:
2049:
2002:
1957:
1897:
1836:
1805:binomial coefficients
1779:
1667:
1614:
1612:{\displaystyle j=r+1}
1580:
1473:
1412:
1349:
1229:
1152:
1150:{\displaystyle B_{3}}
1125:
1101:
1077:
1075:{\displaystyle B_{n}}
1046:
1018:
973:equivalence relations
969:correspond one-to-one
950:
948:{\displaystyle B_{0}}
912:
864:
810:
756:
702:
640:
602:
569:
549:
529:
509:
485:
483:{\displaystyle B_{n}}
458:
442:
407:
387:
385:{\displaystyle B_{n}}
349:
325:
323:{\displaystyle B_{n}}
301:equivalence relations
294:
274:
272:{\displaystyle B_{n}}
233:
156:
102:
82:
80:{\displaystyle B_{n}}
9235:-composition related
9035:Arithmetic functions
8637:Arithmetic functions
8573:Elliptic pseudoprime
8257:Centered icosahedral
8237:Centered tetrahedral
7117:
6163:Touchard polynomials
5716:
5699:{\displaystyle W(n)}
5681:
5654:
5627:
5607:
5581:
5534:
5110:
4958:
4828:
4735:
4636:
4591:
4565:
4541:
4461:
4297:
4230:
4192:
4166:
4113:
4098:; for instance, for
3997:
3910:
3843:Poisson distribution
3737:
3664:
3627:
3597:
3557:
3533:
3455:
3311:
3123:
3095:
3069:
2871:
2843:
2575:
2395:
2274:
2118:
2061:
2057:The Stirling number
1986:
1925:
1814:
1682:
1630:
1591:
1494:
1424:
1382:
1325:permutation patterns
1164:
1157:= 5 factorizations:
1134:
1114:
1090:
1059:
1035:
1007:
932:
926:ordered Bell numbers
874:
820:
766:
712:
652:
611:
578:
558:
538:
518:
498:
467:
414:Poisson distribution
396:
369:
338:
307:
283:
256:
168:
119:
91:
64:
9161:Kaprekar's constant
8681:Colossally abundant
8568:Catalan pseudoprime
8468:Schröder–Hipparchus
8247:Centered octahedral
8123:Centered heptagonal
8113:Centered pentagonal
8103:Centered triangular
7703:and related numbers
7505:1996MaCom..65..383W
7483:Wagstaff, Samuel S.
7467:2008JIntS..11...25S
7248:1978SciAm.238e..24G
7235:Scientific American
7089:The Book of Numbers
7080:Conway, John Horton
6990:2005math......7169C
6357:10.2298/FIL1811881K
6139:Srinivasa Ramanujan
6021:. These are called
6002:was established by
4239:{\displaystyle 173}
3442:combinatorial class
3296:Generating function
3110:{\displaystyle b=k}
3084:{\displaystyle a=1}
2858:{\displaystyle k=1}
1801:recurrence relation
993:equivalence classes
47:partitions of a set
45:count the possible
9579:Mathematics portal
9521:Aronson's sequence
9267:Smarandache–Wellin
9024:-dependent numbers
8731:Primitive abundant
8618:Strong pseudoprime
8608:Perrin pseudoprime
8588:Fermat pseudoprime
8528:Wolstenholme prime
8352:Squared triangular
8138:Centered decagonal
8133:Centered nonagonal
8128:Centered octagonal
8118:Centered hexagonal
7627:Weisstein, Eric W.
7208:Flajolet, Philippe
7147:
7146:
6771:Mathematical Notes
6616:. OEIS Foundation.
6300:. OEIS Foundation.
6100:Dobiński's formula
6084:
5989:
5987:
5696:
5667:
5640:
5613:
5593:
5567:
5517:
5087:
4949:Lambert W function
4937:
4814:
4718:
4619:
4577:
4547:
4527:
4396:
4286:An application of
4236:
4216:
4178:
4162:and for all prime
4149:
4077:
3977:
3887:Modular arithmetic
3809:
3729:Dobinski's formula
3709:
3643:
3641:
3613:
3611:
3583:
3572:
3543:
3516:
3486:
3469:
3411:
3278:
3107:
3081:
3053:
2855:
2827:
2553:
2377:
2251:
2082:
2044:
1952:
1950:
1892:
1795:Summation formulas
1774:
1772:
1662:
1609:
1575:
1468:
1407:
1352:
1224:
1147:
1120:
1096:
1072:
1041:
1013:
945:
922:ordered partitions
907:
859:
805:
751:
697:
635:
597:
564:
544:
524:
504:
480:
461:
453:
435:Partition of a set
402:
382:
344:
320:
289:
269:
228:
151:
97:
77:
9600:Integer sequences
9587:
9586:
9565:
9564:
9534:
9533:
9498:
9497:
9462:
9461:
9426:
9425:
9386:
9385:
9382:
9381:
9201:
9200:
9011:
9010:
8900:
8899:
8896:
8895:
8842:Aliquot sequences
8653:Divisor functions
8626:
8625:
8598:Lucas pseudoprime
8578:Euler pseudoprime
8563:Carmichael number
8541:
8540:
8496:
8495:
8373:
8372:
8369:
8368:
8365:
8364:
8326:
8325:
8214:
8213:
8171:Square triangular
8063:
8062:
8010:
8009:
7962:
7961:
7896:
7895:
7848:
7847:
7786:
7785:
7644:Gottfried Helms.
7212:Sedgewick, Robert
7144:
6411:, pp. 20–23.
6350:(11): 3881–3889.
6119:The Tale of Genji
5974:
5957:
5892:
5855:
5842:
5821:
5746:
5616:{\displaystyle B}
5473:
5347:
5259:
5181:
5065:
5031:
5010:
4984:
4983:
4931:
4833:
4813:
4740:
4706:
4550:{\displaystyle n}
4515:
4384:
4337:
4147:
3990:or, generalizing
3804:
3761:
3370:
3231:
3161:
2991:
2909:
2759:
2693:
2613:
2495:
2433:
2325:
2214:
2194:
2076:
2035:
1943:
1871:
1123:{\displaystyle N}
1099:{\displaystyle N}
1044:{\displaystyle n}
1016:{\displaystyle N}
567:{\displaystyle S}
547:{\displaystyle S}
527:{\displaystyle S}
507:{\displaystyle n}
494:of a set of size
490:is the number of
405:{\displaystyle n}
365:. In particular,
347:{\displaystyle n}
292:{\displaystyle n}
100:{\displaystyle n}
16:(Redirected from
9607:
9577:
9540:
9509:Natural language
9504:
9468:
9436:Generated via a
9432:
9392:
9297:Digit-reassembly
9262:Self-descriptive
9066:
9031:
9017:
8968:Lucas–Carmichael
8958:Harmonic divisor
8906:
8832:Sparsely totient
8807:Highly cototient
8716:Multiply perfect
8706:Highly composite
8649:
8632:
8547:
8502:
8483:Telephone number
8379:
8337:
8318:Square pyramidal
8300:Stella octangula
8225:
8091:
8082:
8074:Figurate numbers
8069:
8016:
7968:
7902:
7854:
7792:
7696:
7680:
7673:
7666:
7657:
7652:
7650:
7640:
7639:
7621:
7604:
7561:
7541:
7531:Wilf, Herbert S.
7526:
7516:
7499:(213): 383–391.
7478:
7452:
7442:
7405:Rota, Gian-Carlo
7399:
7365:
7341:
7309:
7302:Knuth, Donald E.
7297:
7295:
7282:
7259:
7225:
7203:
7172:
7165:Grunert's Archiv
7156:
7154:
7153:
7148:
7145:
7143:
7135:
7134:
7125:
7107:
7075:
7057:
7036:
7026:
7001:
6983:
6958:
6946:
6936:
6926:
6913:
6901:
6888:
6855:
6822:
6789:
6787:
6757:
6731:
6701:
6695:
6689:
6679:
6673:
6663:
6657:
6651:
6645:
6639:
6633:
6627:
6618:
6617:
6599:
6593:
6592:
6590:
6589:
6584:
6575:
6569:
6563:
6557:
6551:
6545:
6539:
6533:
6527:
6521:
6520:
6518:
6517:
6511:
6504:
6499:Complex Analysis
6493:
6487:
6481:
6475:
6469:
6463:
6457:
6451:
6445:
6439:
6433:
6427:
6421:
6412:
6406:
6400:
6394:
6385:
6379:
6370:
6369:
6359:
6335:
6326:
6320:
6314:
6308:
6302:
6301:
6283:
6277:
6271:
6265:
6259:
6253:
6243:
6237:
6236:
6214:Naive set theory
6208:
6202:
6196:
6092:Bell polynomials
6088:Eric Temple Bell
6048:
6034:
5998:
5996:
5995:
5990:
5988:
5975:
5972:
5969:
5966:
5962:
5958:
5956:
5955:
5954:
5932:
5915:
5903:
5902:
5897:
5893:
5891:
5880:
5863:
5856:
5848:
5843:
5841:
5827:
5822:
5820:
5809:
5792:
5747:
5742:
5741:
5740:
5724:
5705:
5703:
5702:
5697:
5676:
5674:
5673:
5668:
5666:
5665:
5649:
5647:
5646:
5641:
5639:
5638:
5622:
5620:
5619:
5614:
5602:
5600:
5599:
5594:
5576:
5574:
5573:
5568:
5526:
5524:
5523:
5518:
5516:
5512:
5508:
5507:
5474:
5472:
5471:
5450:
5449:
5448:
5439:
5438:
5426:
5425:
5416:
5415:
5403:
5402:
5393:
5392:
5380:
5379:
5364:
5363:
5353:
5348:
5346:
5345:
5327:
5326:
5325:
5316:
5315:
5303:
5302:
5287:
5286:
5276:
5260:
5258:
5257:
5256:
5252:
5226:
5216:
5215:
5187:
5182:
5180:
5179:
5178:
5153:
5133:
5128:
5127:
5096:
5094:
5093:
5088:
5083:
5079:
5066:
5064:
5047:
5034:
5033:
5032:
5024:
5015:
5011:
5009:
4992:
4985:
4979:
4975:
4970:
4969:
4946:
4944:
4943:
4938:
4932:
4930:
4919:
4918:
4917:
4898:
4831:
4823:
4821:
4820:
4815:
4811:
4810:
4806:
4796:
4795:
4780:
4779:
4750:
4749:
4738:
4727:
4725:
4724:
4719:
4717:
4716:
4711:
4707:
4705:
4682:
4678:
4677:
4658:
4648:
4647:
4628:
4626:
4625:
4620:
4609:
4608:
4586:
4584:
4583:
4578:
4556:
4554:
4553:
4548:
4536:
4534:
4533:
4528:
4526:
4525:
4520:
4516:
4514:
4491:
4483:
4473:
4472:
4405:
4403:
4402:
4397:
4385:
4383:
4382:
4367:
4366:
4365:
4364:
4350:
4348:
4347:
4338:
4336:
4322:
4314:
4309:
4308:
4272:
4251:
4245:
4243:
4242:
4237:
4225:
4223:
4222:
4217:
4187:
4185:
4184:
4179:
4158:
4156:
4155:
4150:
4148:
4146:
4135:
4128:
4127:
4117:
4086:
4084:
4083:
4078:
4073:
4057:
4056:
4038:
4037:
4022:
4021:
4014:
4013:
3986:
3984:
3983:
3978:
3976:
3960:
3959:
3941:
3940:
3928:
3927:
3818:
3816:
3815:
3810:
3805:
3803:
3795:
3794:
3785:
3782:
3777:
3762:
3754:
3749:
3748:
3718:
3716:
3715:
3710:
3696:
3695:
3674:
3652:
3650:
3649:
3644:
3642:
3622:
3620:
3619:
3614:
3612:
3592:
3590:
3589:
3584:
3582:
3581:
3573:
3552:
3550:
3549:
3544:
3542:
3541:
3525:
3523:
3522:
3517:
3506:
3505:
3496:
3495:
3487:
3470:
3420:
3418:
3417:
3412:
3407:
3406:
3399:
3398:
3381:
3380:
3371:
3369:
3361:
3360:
3351:
3348:
3343:
3287:
3285:
3284:
3279:
3277:
3276:
3252:
3251:
3236:
3232:
3217:
3212:
3194:
3193:
3178:
3177:
3168:
3167:
3166:
3153:
3145:
3140:
3116:
3114:
3113:
3108:
3090:
3088:
3087:
3082:
3062:
3060:
3059:
3054:
3052:
3051:
3036:
3035:
3008:
3007:
2998:
2997:
2996:
2983:
2975:
2970:
2952:
2951:
2942:
2941:
2926:
2925:
2916:
2915:
2914:
2901:
2893:
2888:
2864:
2862:
2861:
2856:
2836:
2834:
2833:
2828:
2823:
2822:
2807:
2806:
2776:
2775:
2766:
2765:
2764:
2751:
2743:
2738:
2723:
2722:
2698:
2694:
2679:
2674:
2656:
2655:
2646:
2645:
2630:
2629:
2620:
2619:
2618:
2605:
2597:
2592:
2562:
2560:
2559:
2554:
2549:
2548:
2527:
2526:
2502:
2501:
2500:
2487:
2479:
2474:
2456:
2455:
2440:
2439:
2438:
2425:
2417:
2412:
2386:
2384:
2383:
2378:
2373:
2372:
2357:
2356:
2332:
2331:
2330:
2317:
2309:
2304:
2286:
2285:
2260:
2258:
2257:
2252:
2247:
2246:
2237:
2236:
2221:
2220:
2219:
2206:
2199:
2195:
2180:
2175:
2159:
2154:
2136:
2135:
2091:
2089:
2088:
2083:
2081:
2077:
2053:
2051:
2050:
2045:
2040:
2036:
2021:
2016:
1998:
1997:
1962:choices for the
1961:
1959:
1958:
1953:
1951:
1949:
1948:
1935:
1901:
1899:
1898:
1893:
1888:
1887:
1878:
1877:
1876:
1863:
1855:
1850:
1832:
1831:
1783:
1781:
1780:
1775:
1773:
1671:
1669:
1668:
1663:
1661:
1660:
1642:
1641:
1618:
1616:
1615:
1610:
1584:
1582:
1581:
1576:
1571:
1570:
1540:
1539:
1515:
1514:
1477:
1475:
1474:
1469:
1467:
1466:
1442:
1441:
1416:
1414:
1413:
1408:
1400:
1399:
1368:Alexander Aitken
1233:
1231:
1230:
1225:
1156:
1154:
1153:
1148:
1146:
1145:
1129:
1127:
1126:
1121:
1105:
1103:
1102:
1097:
1081:
1079:
1078:
1073:
1071:
1070:
1050:
1048:
1047:
1042:
1022:
1020:
1019:
1014:
977:binary relations
954:
952:
951:
946:
944:
943:
916:
914:
913:
908:
868:
866:
865:
860:
814:
812:
811:
806:
760:
758:
757:
752:
706:
704:
703:
698:
644:
642:
641:
636:
606:
604:
603:
598:
590:
589:
573:
571:
570:
565:
553:
551:
550:
545:
533:
531:
530:
525:
513:
511:
510:
505:
489:
487:
486:
481:
479:
478:
412:-th moment of a
411:
409:
408:
403:
391:
389:
388:
383:
381:
380:
353:
351:
350:
345:
329:
327:
326:
321:
319:
318:
298:
296:
295:
290:
278:
276:
275:
270:
268:
267:
252:The Bell number
243:
237:
235:
234:
229:
160:
158:
157:
152:
144:
143:
131:
130:
115:. Starting with
106:
104:
103:
98:
86:
84:
83:
78:
76:
75:
55:Eric Temple Bell
21:
9615:
9614:
9610:
9609:
9608:
9606:
9605:
9604:
9590:
9589:
9588:
9583:
9561:
9557:Strobogrammatic
9548:
9530:
9512:
9494:
9476:
9458:
9440:
9422:
9399:
9378:
9362:
9321:Divisor-related
9316:
9276:
9227:
9197:
9134:
9118:
9097:
9064:
9037:
9025:
9007:
8919:
8918:related numbers
8892:
8869:
8836:
8827:Perfect totient
8793:
8770:
8701:Highly abundant
8643:
8622:
8554:
8537:
8509:
8492:
8478:Stirling second
8384:
8361:
8322:
8304:
8261:
8210:
8147:
8108:Centered square
8076:
8059:
8021:
8006:
7973:
7958:
7910:
7909:defined numbers
7892:
7859:
7844:
7815:Double Mersenne
7801:
7782:
7704:
7690:
7688:natural numbers
7684:
7648:
7643:
7625:
7624:
7616:Robert Dickau.
7615:
7612:
7607:
7585:10.2307/2305292
7564:
7550:
7539:
7529:
7481:
7450:
7445:
7423:10.2307/2312585
7403:
7388:10.2307/2369442
7368:
7344:
7330:
7312:
7300:
7285:
7267:
7230:Gardner, Martin
7228:
7206:
7175:
7136:
7126:
7115:
7114:
7110:
7104:
7084:Guy, Richard K.
7078:
7039:
7004:
6961:
6951:de Bruijn, N.G.
6949:
6934:
6929:
6916:
6910:
6899:
6892:
6877:10.2307/1968633
6859:
6844:10.2307/1968431
6826:
6811:10.2307/2372336
6793:Becker, H. W.;
6792:
6760:
6713:
6709:
6704:
6696:
6692:
6680:
6676:
6664:
6660:
6652:
6648:
6640:
6636:
6628:
6621:
6601:
6600:
6596:
6587:
6585:
6582:
6577:
6576:
6572:
6564:
6560:
6552:
6548:
6540:
6536:
6528:
6524:
6515:
6513:
6509:
6502:
6495:
6494:
6490:
6482:
6478:
6470:
6466:
6458:
6454:
6446:
6442:
6434:
6430:
6422:
6415:
6407:
6403:
6395:
6388:
6380:
6373:
6337:
6336:
6329:
6321:
6317:
6309:
6305:
6285:
6284:
6280:
6272:
6268:
6262:Claesson (2001)
6260:
6256:
6244:
6240:
6225:
6210:
6209:
6205:
6197:
6190:
6186:
6173:Stirling number
6159:
6133:
6107:
6070:
6063:
6044:
6030:
6012:
5986:
5985:
5964:
5963:
5946:
5933:
5916:
5910:
5881:
5864:
5858:
5857:
5831:
5810:
5793:
5748:
5732:
5725:
5714:
5713:
5679:
5678:
5657:
5652:
5651:
5630:
5625:
5624:
5605:
5604:
5579:
5578:
5532:
5531:
5484:
5451:
5440:
5430:
5417:
5407:
5394:
5384:
5371:
5355:
5354:
5328:
5317:
5307:
5294:
5278:
5277:
5268:
5264:
5240:
5227:
5198:
5188:
5164:
5154:
5134:
5113:
5108:
5107:
5051:
5045:
5041:
4996:
4987:
4986:
4961:
4956:
4955:
4920:
4906:
4899:
4826:
4825:
4784:
4771:
4770:
4766:
4741:
4733:
4732:
4683:
4660:
4659:
4653:
4652:
4639:
4634:
4633:
4600:
4589:
4588:
4563:
4562:
4539:
4538:
4492:
4484:
4478:
4477:
4464:
4459:
4458:
4444:
4433:
4419:
4368:
4356:
4351:
4339:
4323:
4315:
4300:
4295:
4294:
4284:
4268:
4247:
4228:
4227:
4190:
4189:
4164:
4163:
4136:
4119:
4118:
4111:
4110:
4042:
4029:
4005:
4000:
3995:
3994:
3945:
3932:
3913:
3908:
3907:
3889:
3831:
3796:
3786:
3740:
3735:
3734:
3725:
3687:
3667:
3662:
3661:
3625:
3624:
3595:
3594:
3560:
3555:
3554:
3531:
3530:
3474:
3453:
3452:
3390:
3385:
3372:
3362:
3352:
3309:
3308:
3298:
3262:
3237:
3219:
3179:
3169:
3148:
3121:
3120:
3093:
3092:
3067:
3066:
3037:
3021:
2999:
2978:
2943:
2927:
2917:
2896:
2869:
2868:
2841:
2840:
2808:
2792:
2767:
2746:
2708:
2681:
2647:
2631:
2621:
2600:
2573:
2572:
2528:
2512:
2482:
2441:
2420:
2393:
2392:
2358:
2342:
2312:
2277:
2272:
2271:
2238:
2222:
2201:
2182:
2121:
2116:
2115:
2064:
2059:
2058:
2023:
1989:
1984:
1983:
1971:
1930:
1923:
1922:
1879:
1858:
1817:
1812:
1811:
1797:
1792:
1771:
1770:
1765:
1760:
1755:
1750:
1744:
1743:
1738:
1733:
1728:
1722:
1721:
1716:
1711:
1705:
1704:
1699:
1693:
1692:
1680:
1679:
1646:
1633:
1628:
1627:
1626:for that row. (
1589:
1588:
1544:
1519:
1500:
1492:
1491:
1446:
1427:
1422:
1421:
1385:
1380:
1379:
1364:Peirce triangle
1344:
1338:
1298:
1291:
1274:. If a deck of
1264:
1240:
1162:
1161:
1137:
1132:
1131:
1112:
1111:
1088:
1087:
1062:
1057:
1056:
1033:
1032:
1005:
1004:
1001:
935:
930:
929:
872:
871:
818:
817:
764:
763:
710:
709:
650:
649:
609:
608:
581:
576:
575:
574:. For example,
556:
555:
554:whose union is
536:
535:
516:
515:
496:
495:
470:
465:
464:
451: − 1.
437:
431:
426:
394:
393:
372:
367:
366:
336:
335:
310:
305:
304:
281:
280:
259:
254:
253:
239:
166:
165:
135:
122:
117:
116:
89:
88:
67:
62:
61:
35:
28:
23:
22:
15:
12:
11:
5:
9613:
9611:
9603:
9602:
9592:
9591:
9585:
9584:
9582:
9581:
9570:
9567:
9566:
9563:
9562:
9560:
9559:
9553:
9550:
9549:
9543:
9536:
9535:
9532:
9531:
9529:
9528:
9523:
9517:
9514:
9513:
9507:
9500:
9499:
9496:
9495:
9493:
9492:
9490:Sorting number
9487:
9485:Pancake number
9481:
9478:
9477:
9471:
9464:
9463:
9460:
9459:
9457:
9456:
9451:
9445:
9442:
9441:
9435:
9428:
9427:
9424:
9423:
9421:
9420:
9415:
9410:
9404:
9401:
9400:
9397:Binary numbers
9395:
9388:
9387:
9384:
9383:
9380:
9379:
9377:
9376:
9370:
9368:
9364:
9363:
9361:
9360:
9355:
9350:
9345:
9340:
9335:
9330:
9324:
9322:
9318:
9317:
9315:
9314:
9309:
9304:
9299:
9294:
9288:
9286:
9278:
9277:
9275:
9274:
9269:
9264:
9259:
9254:
9249:
9244:
9238:
9236:
9229:
9228:
9226:
9225:
9224:
9223:
9212:
9210:
9207:P-adic numbers
9203:
9202:
9199:
9198:
9196:
9195:
9194:
9193:
9183:
9178:
9173:
9168:
9163:
9158:
9153:
9148:
9142:
9140:
9136:
9135:
9133:
9132:
9126:
9124:
9123:Coding-related
9120:
9119:
9117:
9116:
9111:
9105:
9103:
9099:
9098:
9096:
9095:
9090:
9085:
9080:
9074:
9072:
9063:
9062:
9061:
9060:
9058:Multiplicative
9055:
9044:
9042:
9027:
9026:
9022:Numeral system
9020:
9013:
9012:
9009:
9008:
9006:
9005:
9000:
8995:
8990:
8985:
8980:
8975:
8970:
8965:
8960:
8955:
8950:
8945:
8940:
8935:
8930:
8924:
8921:
8920:
8909:
8902:
8901:
8898:
8897:
8894:
8893:
8891:
8890:
8885:
8879:
8877:
8871:
8870:
8868:
8867:
8862:
8857:
8852:
8846:
8844:
8838:
8837:
8835:
8834:
8829:
8824:
8819:
8814:
8812:Highly totient
8809:
8803:
8801:
8795:
8794:
8792:
8791:
8786:
8780:
8778:
8772:
8771:
8769:
8768:
8763:
8758:
8753:
8748:
8743:
8738:
8733:
8728:
8723:
8718:
8713:
8708:
8703:
8698:
8693:
8688:
8683:
8678:
8673:
8668:
8666:Almost perfect
8663:
8657:
8655:
8645:
8644:
8635:
8628:
8627:
8624:
8623:
8621:
8620:
8615:
8610:
8605:
8600:
8595:
8590:
8585:
8580:
8575:
8570:
8565:
8559:
8556:
8555:
8550:
8543:
8542:
8539:
8538:
8536:
8535:
8530:
8525:
8520:
8514:
8511:
8510:
8505:
8498:
8497:
8494:
8493:
8491:
8490:
8485:
8480:
8475:
8473:Stirling first
8470:
8465:
8460:
8455:
8450:
8445:
8440:
8435:
8430:
8425:
8420:
8415:
8410:
8405:
8400:
8395:
8389:
8386:
8385:
8382:
8375:
8374:
8371:
8370:
8367:
8366:
8363:
8362:
8360:
8359:
8354:
8349:
8343:
8341:
8334:
8328:
8327:
8324:
8323:
8321:
8320:
8314:
8312:
8306:
8305:
8303:
8302:
8297:
8292:
8287:
8282:
8277:
8271:
8269:
8263:
8262:
8260:
8259:
8254:
8249:
8244:
8239:
8233:
8231:
8222:
8216:
8215:
8212:
8211:
8209:
8208:
8203:
8198:
8193:
8188:
8183:
8178:
8173:
8168:
8163:
8157:
8155:
8149:
8148:
8146:
8145:
8140:
8135:
8130:
8125:
8120:
8115:
8110:
8105:
8099:
8097:
8088:
8078:
8077:
8072:
8065:
8064:
8061:
8060:
8058:
8057:
8052:
8047:
8042:
8037:
8032:
8026:
8023:
8022:
8019:
8012:
8011:
8008:
8007:
8005:
8004:
7999:
7994:
7989:
7984:
7978:
7975:
7974:
7971:
7964:
7963:
7960:
7959:
7957:
7956:
7951:
7946:
7941:
7936:
7931:
7926:
7921:
7915:
7912:
7911:
7905:
7898:
7897:
7894:
7893:
7891:
7890:
7885:
7880:
7875:
7870:
7864:
7861:
7860:
7857:
7850:
7849:
7846:
7845:
7843:
7842:
7837:
7832:
7827:
7822:
7817:
7812:
7806:
7803:
7802:
7795:
7788:
7787:
7784:
7783:
7781:
7780:
7775:
7770:
7765:
7760:
7755:
7750:
7745:
7740:
7735:
7730:
7725:
7720:
7715:
7709:
7706:
7705:
7699:
7692:
7691:
7685:
7683:
7682:
7675:
7668:
7660:
7654:
7653:
7641:
7622:
7611:
7610:External links
7608:
7606:
7605:
7562:
7548:
7527:
7479:
7443:
7417:(5): 498–504.
7401:
7366:
7342:
7328:
7310:
7298:
7283:
7269:"Bell numbers"
7265:
7226:
7204:
7173:
7142:
7139:
7133:
7129:
7123:
7108:
7102:
7076:
7048:(7): 961–971.
7037:
7017:(1): 184–187.
7002:
6959:
6947:
6927:
6914:
6908:
6890:
6871:(3): 539–557.
6857:
6838:(2): 258–277.
6824:
6805:(2): 385–394.
6790:
6758:
6722:(1–3): 79–87.
6710:
6708:
6705:
6703:
6702:
6690:
6674:
6658:
6646:
6634:
6619:
6594:
6570:
6558:
6546:
6534:
6522:
6488:
6476:
6464:
6452:
6440:
6428:
6413:
6401:
6386:
6371:
6327:
6315:
6303:
6278:
6266:
6254:
6238:
6223:
6203:
6187:
6185:
6182:
6181:
6180:
6175:
6170:
6168:Catalan number
6165:
6158:
6155:
6131:
6105:
6069:
6066:
6061:
6041:
6040:
6011:
6008:
6004:de Bruijn 1981
6000:
5999:
5984:
5981:
5978:
5967:
5965:
5961:
5953:
5949:
5945:
5942:
5939:
5936:
5931:
5928:
5925:
5922:
5919:
5913:
5909:
5906:
5901:
5896:
5890:
5887:
5884:
5879:
5876:
5873:
5870:
5867:
5861:
5854:
5851:
5846:
5840:
5837:
5834:
5830:
5825:
5819:
5816:
5813:
5808:
5805:
5802:
5799:
5796:
5790:
5787:
5784:
5781:
5778:
5775:
5772:
5769:
5766:
5763:
5760:
5757:
5754:
5751:
5749:
5745:
5739:
5735:
5731:
5728:
5722:
5721:
5695:
5692:
5689:
5686:
5664:
5660:
5637:
5633:
5612:
5592:
5589:
5586:
5566:
5563:
5560:
5557:
5554:
5551:
5548:
5545:
5542:
5539:
5530:uniformly for
5528:
5527:
5515:
5511:
5506:
5503:
5500:
5497:
5494:
5491:
5487:
5483:
5480:
5477:
5470:
5467:
5464:
5461:
5458:
5454:
5447:
5443:
5437:
5433:
5429:
5424:
5420:
5414:
5410:
5406:
5401:
5397:
5391:
5387:
5383:
5378:
5374:
5370:
5367:
5362:
5358:
5351:
5344:
5341:
5338:
5335:
5331:
5324:
5320:
5314:
5310:
5306:
5301:
5297:
5293:
5290:
5285:
5281:
5274:
5271:
5267:
5263:
5255:
5251:
5247:
5243:
5239:
5236:
5233:
5230:
5225:
5222:
5219:
5214:
5211:
5208:
5205:
5201:
5197:
5194:
5191:
5185:
5177:
5174:
5171:
5167:
5163:
5160:
5157:
5152:
5149:
5146:
5143:
5140:
5137:
5131:
5126:
5123:
5120:
5116:
5098:
5097:
5086:
5082:
5078:
5075:
5072:
5069:
5063:
5060:
5057:
5054:
5050:
5044:
5040:
5037:
5030:
5027:
5022:
5019:
5014:
5008:
5005:
5002:
4999:
4995:
4990:
4982:
4978:
4973:
4968:
4964:
4936:
4929:
4926:
4923:
4916:
4913:
4909:
4905:
4902:
4896:
4893:
4890:
4887:
4884:
4881:
4878:
4875:
4872:
4869:
4866:
4863:
4860:
4857:
4854:
4851:
4848:
4845:
4842:
4839:
4836:
4809:
4805:
4802:
4799:
4794:
4791:
4787:
4783:
4778:
4774:
4769:
4765:
4762:
4759:
4756:
4753:
4748:
4744:
4729:
4728:
4715:
4710:
4704:
4701:
4698:
4695:
4692:
4689:
4686:
4681:
4676:
4673:
4670:
4667:
4663:
4656:
4651:
4646:
4642:
4618:
4615:
4612:
4607:
4603:
4599:
4596:
4576:
4573:
4570:
4559:
4558:
4546:
4524:
4519:
4513:
4510:
4507:
4504:
4501:
4498:
4495:
4490:
4487:
4481:
4476:
4471:
4467:
4443:
4440:
4429:
4418:
4415:
4407:
4406:
4395:
4392:
4389:
4381:
4378:
4375:
4371:
4363:
4359:
4354:
4346:
4342:
4335:
4332:
4329:
4326:
4321:
4318:
4312:
4307:
4303:
4283:
4280:
4279:
4278:
4235:
4215:
4212:
4209:
4206:
4203:
4200:
4197:
4177:
4174:
4171:
4160:
4159:
4145:
4142:
4139:
4134:
4131:
4126:
4122:
4088:
4087:
4076:
4072:
4069:
4065:
4062:
4055:
4052:
4049:
4045:
4041:
4036:
4032:
4028:
4025:
4020:
4017:
4012:
4008:
4003:
3988:
3987:
3975:
3972:
3968:
3965:
3958:
3955:
3952:
3948:
3944:
3939:
3935:
3931:
3926:
3923:
3920:
3916:
3888:
3885:
3847:expected value
3829:
3820:
3819:
3808:
3802:
3799:
3793:
3789:
3781:
3776:
3773:
3770:
3766:
3760:
3757:
3752:
3747:
3743:
3724:
3721:
3708:
3705:
3702:
3699:
3694:
3690:
3686:
3683:
3680:
3677:
3673:
3670:
3640:
3637:
3633:
3610:
3607:
3603:
3580:
3577:
3571:
3568:
3564:
3540:
3527:
3526:
3515:
3512:
3509:
3504:
3499:
3494:
3491:
3485:
3482:
3478:
3473:
3468:
3465:
3461:
3422:
3421:
3410:
3405:
3402:
3397:
3393:
3388:
3384:
3379:
3375:
3368:
3365:
3359:
3355:
3347:
3342:
3339:
3336:
3332:
3328:
3325:
3322:
3319:
3316:
3297:
3294:
3275:
3272:
3269:
3265:
3261:
3258:
3255:
3250:
3247:
3244:
3240:
3235:
3230:
3227:
3222:
3216:
3211:
3208:
3205:
3201:
3197:
3192:
3189:
3186:
3182:
3176:
3172:
3165:
3160:
3157:
3152:
3144:
3139:
3136:
3133:
3129:
3106:
3103:
3100:
3080:
3077:
3074:
3050:
3047:
3044:
3040:
3034:
3031:
3028:
3024:
3020:
3017:
3014:
3011:
3006:
3002:
2995:
2990:
2987:
2982:
2974:
2969:
2966:
2963:
2959:
2955:
2950:
2946:
2940:
2937:
2934:
2930:
2924:
2920:
2913:
2908:
2905:
2900:
2892:
2887:
2884:
2881:
2877:
2854:
2851:
2848:
2826:
2821:
2818:
2815:
2811:
2805:
2802:
2799:
2795:
2791:
2788:
2785:
2782:
2779:
2774:
2770:
2763:
2758:
2755:
2750:
2742:
2737:
2734:
2731:
2727:
2721:
2718:
2715:
2711:
2707:
2704:
2701:
2697:
2692:
2689:
2684:
2678:
2673:
2670:
2667:
2663:
2659:
2654:
2650:
2644:
2641:
2638:
2634:
2628:
2624:
2617:
2612:
2609:
2604:
2596:
2591:
2588:
2585:
2581:
2552:
2547:
2544:
2541:
2538:
2535:
2531:
2525:
2522:
2519:
2515:
2511:
2508:
2505:
2499:
2494:
2491:
2486:
2478:
2473:
2470:
2467:
2463:
2459:
2454:
2451:
2448:
2444:
2437:
2432:
2429:
2424:
2416:
2411:
2408:
2405:
2401:
2376:
2371:
2368:
2365:
2361:
2355:
2352:
2349:
2345:
2341:
2338:
2335:
2329:
2324:
2321:
2316:
2308:
2303:
2300:
2297:
2293:
2289:
2284:
2280:
2262:
2261:
2250:
2245:
2241:
2235:
2232:
2229:
2225:
2218:
2213:
2210:
2205:
2198:
2193:
2190:
2185:
2179:
2174:
2171:
2168:
2164:
2158:
2153:
2150:
2147:
2143:
2139:
2134:
2131:
2128:
2124:
2080:
2075:
2072:
2067:
2055:
2054:
2043:
2039:
2034:
2031:
2026:
2020:
2015:
2012:
2009:
2005:
2001:
1996:
1992:
1969:
1947:
1942:
1939:
1934:
1903:
1902:
1891:
1886:
1882:
1875:
1870:
1867:
1862:
1854:
1849:
1846:
1843:
1839:
1835:
1830:
1827:
1824:
1820:
1796:
1793:
1791:
1788:
1769:
1766:
1764:
1761:
1759:
1756:
1754:
1751:
1749:
1746:
1745:
1742:
1739:
1737:
1734:
1732:
1729:
1727:
1724:
1723:
1720:
1717:
1715:
1712:
1710:
1707:
1706:
1703:
1700:
1698:
1695:
1694:
1691:
1688:
1687:
1674:
1673:
1659:
1656:
1653:
1649:
1645:
1640:
1636:
1620:
1608:
1605:
1602:
1599:
1596:
1585:
1574:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1547:
1543:
1538:
1535:
1532:
1529:
1526:
1522:
1518:
1513:
1510:
1507:
1503:
1499:
1487:
1465:
1462:
1459:
1456:
1453:
1449:
1445:
1440:
1437:
1434:
1430:
1418:
1406:
1403:
1398:
1395:
1392:
1388:
1360:Aitken's array
1358:, also called
1340:Main article:
1337:
1334:
1296:
1289:
1263:
1260:
1239:
1236:
1235:
1234:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1144:
1140:
1119:
1108:factorizations
1095:
1069:
1065:
1040:
1012:
1000:
999:Factorizations
997:
961:vacuously true
942:
938:
918:
917:
906:
903:
900:
897:
894:
891:
888:
885:
882:
879:
869:
858:
855:
852:
849:
846:
843:
840:
837:
834:
831:
828:
825:
815:
804:
801:
798:
795:
792:
789:
786:
783:
780:
777:
774:
771:
761:
750:
747:
744:
741:
738:
735:
732:
729:
726:
723:
720:
717:
707:
696:
693:
690:
687:
684:
681:
678:
675:
672:
669:
666:
663:
660:
657:
634:
631:
628:
625:
622:
619:
616:
596:
593:
588:
584:
563:
543:
523:
503:
477:
473:
433:Main article:
430:
429:Set partitions
427:
425:
422:
401:
379:
375:
343:
317:
313:
288:
266:
262:
250:
249:
227:
224:
221:
218:
215:
212:
209:
206:
203:
200:
197:
194:
191:
188:
185:
182:
179:
176:
173:
150:
147:
142:
138:
134:
129:
125:
96:
74:
70:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
9612:
9601:
9598:
9597:
9595:
9580:
9576:
9572:
9571:
9568:
9558:
9555:
9554:
9551:
9546:
9541:
9537:
9527:
9524:
9522:
9519:
9518:
9515:
9510:
9505:
9501:
9491:
9488:
9486:
9483:
9482:
9479:
9474:
9469:
9465:
9455:
9452:
9450:
9447:
9446:
9443:
9439:
9433:
9429:
9419:
9416:
9414:
9411:
9409:
9406:
9405:
9402:
9398:
9393:
9389:
9375:
9372:
9371:
9369:
9365:
9359:
9356:
9354:
9351:
9349:
9348:Polydivisible
9346:
9344:
9341:
9339:
9336:
9334:
9331:
9329:
9326:
9325:
9323:
9319:
9313:
9310:
9308:
9305:
9303:
9300:
9298:
9295:
9293:
9290:
9289:
9287:
9284:
9279:
9273:
9270:
9268:
9265:
9263:
9260:
9258:
9255:
9253:
9250:
9248:
9245:
9243:
9240:
9239:
9237:
9234:
9230:
9222:
9219:
9218:
9217:
9214:
9213:
9211:
9208:
9204:
9192:
9189:
9188:
9187:
9184:
9182:
9179:
9177:
9174:
9172:
9169:
9167:
9164:
9162:
9159:
9157:
9154:
9152:
9149:
9147:
9144:
9143:
9141:
9137:
9131:
9128:
9127:
9125:
9121:
9115:
9112:
9110:
9107:
9106:
9104:
9102:Digit product
9100:
9094:
9091:
9089:
9086:
9084:
9081:
9079:
9076:
9075:
9073:
9071:
9067:
9059:
9056:
9054:
9051:
9050:
9049:
9046:
9045:
9043:
9041:
9036:
9032:
9028:
9023:
9018:
9014:
9004:
9001:
8999:
8996:
8994:
8991:
8989:
8986:
8984:
8981:
8979:
8976:
8974:
8971:
8969:
8966:
8964:
8961:
8959:
8956:
8954:
8951:
8949:
8946:
8944:
8941:
8939:
8938:Erdős–Nicolas
8936:
8934:
8931:
8929:
8926:
8925:
8922:
8917:
8913:
8907:
8903:
8889:
8886:
8884:
8881:
8880:
8878:
8876:
8872:
8866:
8863:
8861:
8858:
8856:
8853:
8851:
8848:
8847:
8845:
8843:
8839:
8833:
8830:
8828:
8825:
8823:
8820:
8818:
8815:
8813:
8810:
8808:
8805:
8804:
8802:
8800:
8796:
8790:
8787:
8785:
8782:
8781:
8779:
8777:
8773:
8767:
8764:
8762:
8759:
8757:
8756:Superabundant
8754:
8752:
8749:
8747:
8744:
8742:
8739:
8737:
8734:
8732:
8729:
8727:
8724:
8722:
8719:
8717:
8714:
8712:
8709:
8707:
8704:
8702:
8699:
8697:
8694:
8692:
8689:
8687:
8684:
8682:
8679:
8677:
8674:
8672:
8669:
8667:
8664:
8662:
8659:
8658:
8656:
8654:
8650:
8646:
8642:
8638:
8633:
8629:
8619:
8616:
8614:
8611:
8609:
8606:
8604:
8601:
8599:
8596:
8594:
8591:
8589:
8586:
8584:
8581:
8579:
8576:
8574:
8571:
8569:
8566:
8564:
8561:
8560:
8557:
8553:
8548:
8544:
8534:
8531:
8529:
8526:
8524:
8521:
8519:
8516:
8515:
8512:
8508:
8503:
8499:
8489:
8486:
8484:
8481:
8479:
8476:
8474:
8471:
8469:
8466:
8464:
8461:
8459:
8456:
8454:
8451:
8449:
8446:
8444:
8441:
8439:
8436:
8434:
8431:
8429:
8426:
8424:
8421:
8419:
8416:
8414:
8411:
8409:
8406:
8404:
8401:
8399:
8396:
8394:
8391:
8390:
8387:
8380:
8376:
8358:
8355:
8353:
8350:
8348:
8345:
8344:
8342:
8338:
8335:
8333:
8332:4-dimensional
8329:
8319:
8316:
8315:
8313:
8311:
8307:
8301:
8298:
8296:
8293:
8291:
8288:
8286:
8283:
8281:
8278:
8276:
8273:
8272:
8270:
8268:
8264:
8258:
8255:
8253:
8250:
8248:
8245:
8243:
8242:Centered cube
8240:
8238:
8235:
8234:
8232:
8230:
8226:
8223:
8221:
8220:3-dimensional
8217:
8207:
8204:
8202:
8199:
8197:
8194:
8192:
8189:
8187:
8184:
8182:
8179:
8177:
8174:
8172:
8169:
8167:
8164:
8162:
8159:
8158:
8156:
8154:
8150:
8144:
8141:
8139:
8136:
8134:
8131:
8129:
8126:
8124:
8121:
8119:
8116:
8114:
8111:
8109:
8106:
8104:
8101:
8100:
8098:
8096:
8092:
8089:
8087:
8086:2-dimensional
8083:
8079:
8075:
8070:
8066:
8056:
8053:
8051:
8048:
8046:
8043:
8041:
8038:
8036:
8033:
8031:
8030:Nonhypotenuse
8028:
8027:
8024:
8017:
8013:
8003:
8000:
7998:
7995:
7993:
7990:
7988:
7985:
7983:
7980:
7979:
7976:
7969:
7965:
7955:
7952:
7950:
7947:
7945:
7942:
7940:
7937:
7935:
7932:
7930:
7927:
7925:
7922:
7920:
7917:
7916:
7913:
7908:
7903:
7899:
7889:
7886:
7884:
7881:
7879:
7876:
7874:
7871:
7869:
7866:
7865:
7862:
7855:
7851:
7841:
7838:
7836:
7833:
7831:
7828:
7826:
7823:
7821:
7818:
7816:
7813:
7811:
7808:
7807:
7804:
7799:
7793:
7789:
7779:
7776:
7774:
7771:
7769:
7768:Perfect power
7766:
7764:
7761:
7759:
7758:Seventh power
7756:
7754:
7751:
7749:
7746:
7744:
7741:
7739:
7736:
7734:
7731:
7729:
7726:
7724:
7721:
7719:
7716:
7714:
7711:
7710:
7707:
7702:
7697:
7693:
7689:
7681:
7676:
7674:
7669:
7667:
7662:
7661:
7658:
7647:
7642:
7637:
7636:
7631:
7630:"Bell Number"
7628:
7623:
7619:
7614:
7613:
7609:
7602:
7598:
7594:
7590:
7586:
7582:
7578:
7574:
7573:
7568:
7563:
7559:
7555:
7551:
7549:0-12-751956-4
7545:
7538:
7537:
7532:
7528:
7524:
7520:
7515:
7510:
7506:
7502:
7498:
7494:
7493:
7488:
7484:
7480:
7476:
7472:
7468:
7464:
7460:
7456:
7449:
7444:
7440:
7436:
7432:
7428:
7424:
7420:
7416:
7412:
7411:
7406:
7402:
7397:
7393:
7389:
7385:
7381:
7377:
7376:
7371:
7370:Peirce, C. S.
7367:
7363:
7359:
7355:
7351:
7347:
7343:
7339:
7335:
7331:
7329:9780821869475
7325:
7321:
7320:
7315:
7311:
7307:
7303:
7299:
7294:
7289:
7284:
7280:
7276:
7275:
7270:
7266:
7263:
7257:
7253:
7249:
7245:
7241:
7237:
7236:
7231:
7227:
7223:
7219:
7218:
7213:
7209:
7205:
7201:
7197:
7193:
7189:
7185:
7181:
7180:
7174:
7170:
7166:
7162:
7160:
7140:
7137:
7131:
7127:
7121:
7109:
7105:
7103:9780387979939
7099:
7095:
7091:
7090:
7085:
7081:
7077:
7073:
7069:
7065:
7061:
7056:
7051:
7047:
7043:
7038:
7034:
7030:
7025:
7020:
7016:
7012:
7008:
7003:
6999:
6995:
6991:
6987:
6982:
6977:
6974:(1): 06.1.4.
6973:
6969:
6965:
6960:
6956:
6952:
6948:
6944:
6940:
6933:
6928:
6925:(2): 185–205.
6924:
6920:
6915:
6911:
6909:0-486-44603-4
6905:
6898:
6897:
6891:
6886:
6882:
6878:
6874:
6870:
6866:
6862:
6858:
6853:
6849:
6845:
6841:
6837:
6833:
6829:
6825:
6820:
6816:
6812:
6808:
6804:
6800:
6796:
6795:Riordan, John
6791:
6786:
6781:
6777:
6773:
6772:
6767:
6763:
6762:Aitken, A. C.
6759:
6755:
6751:
6747:
6743:
6739:
6735:
6730:
6725:
6721:
6717:
6712:
6711:
6706:
6699:
6694:
6691:
6687:
6683:
6678:
6675:
6671:
6667:
6662:
6659:
6655:
6650:
6647:
6643:
6638:
6635:
6631:
6626:
6624:
6620:
6615:
6614:
6608:
6604:
6598:
6595:
6581:
6574:
6571:
6567:
6566:Lovász (1993)
6562:
6559:
6555:
6550:
6547:
6543:
6542:Canfield 1995
6538:
6535:
6531:
6526:
6523:
6512:on 2014-01-24
6508:
6501:
6500:
6492:
6489:
6485:
6484:Wagstaff 1996
6480:
6477:
6473:
6472:Williams 1945
6468:
6465:
6461:
6456:
6453:
6449:
6444:
6441:
6437:
6436:Dobiński 1877
6432:
6429:
6425:
6420:
6418:
6414:
6410:
6405:
6402:
6398:
6393:
6391:
6387:
6383:
6378:
6376:
6372:
6367:
6363:
6358:
6353:
6349:
6345:
6341:
6334:
6332:
6328:
6324:
6319:
6316:
6313:, p. 23.
6312:
6307:
6304:
6299:
6298:
6292:
6288:
6282:
6279:
6275:
6274:Callan (2006)
6270:
6267:
6263:
6258:
6255:
6251:
6247:
6246:Williams 1945
6242:
6239:
6234:
6230:
6226:
6224:9781475716450
6220:
6216:
6215:
6207:
6204:
6200:
6195:
6193:
6189:
6183:
6179:
6176:
6174:
6171:
6169:
6166:
6164:
6161:
6160:
6156:
6154:
6152:
6148:
6144:
6143:Bell triangle
6140:
6135:
6130:
6125:
6121:
6120:
6114:
6112:
6108:
6101:
6097:
6096:Dobiński 1877
6093:
6089:
6081:
6080:
6079:Tale of Genji
6074:
6067:
6065:
6060:
6056:
6052:
6047:
6038:
6033:
6028:
6027:
6026:
6024:
6020:
6019:prime numbers
6016:
6009:
6007:
6005:
5976:
5959:
5951:
5943:
5940:
5937:
5929:
5926:
5923:
5920:
5917:
5911:
5907:
5904:
5899:
5894:
5888:
5885:
5882:
5877:
5874:
5871:
5868:
5865:
5859:
5852:
5849:
5844:
5838:
5835:
5832:
5828:
5823:
5817:
5814:
5811:
5806:
5803:
5800:
5797:
5794:
5788:
5785:
5782:
5779:
5776:
5773:
5770:
5767:
5764:
5761:
5758:
5755:
5752:
5750:
5743:
5737:
5733:
5729:
5726:
5712:
5711:
5710:
5707:
5690:
5684:
5662:
5658:
5635:
5631:
5610:
5584:
5558:
5552:
5549:
5543:
5540:
5537:
5513:
5501:
5495:
5492:
5489:
5485:
5478:
5475:
5465:
5459:
5456:
5452:
5445:
5441:
5435:
5431:
5427:
5422:
5418:
5412:
5408:
5404:
5399:
5395:
5389:
5385:
5381:
5376:
5372:
5368:
5365:
5360:
5356:
5349:
5339:
5333:
5329:
5322:
5318:
5312:
5308:
5304:
5299:
5295:
5291:
5288:
5283:
5279:
5272:
5269:
5265:
5261:
5253:
5249:
5245:
5237:
5234:
5231:
5220:
5217:
5209:
5203:
5199:
5192:
5189:
5183:
5175:
5172:
5169:
5161:
5155:
5150:
5144:
5141:
5138:
5129:
5124:
5121:
5118:
5114:
5106:
5105:
5104:
5102:
5084:
5080:
5076:
5073:
5070:
5067:
5058:
5052:
5048:
5042:
5038:
5035:
5028:
5025:
5020:
5017:
5012:
5003:
4997:
4993:
4988:
4980:
4976:
4971:
4966:
4962:
4954:
4953:
4952:
4950:
4934:
4927:
4924:
4921:
4914:
4911:
4907:
4903:
4900:
4894:
4891:
4888:
4885:
4882:
4879:
4876:
4870:
4867:
4864:
4858:
4855:
4852:
4849:
4846:
4840:
4834:
4807:
4800:
4792:
4789:
4785:
4781:
4776:
4772:
4767:
4760:
4754:
4746:
4742:
4713:
4708:
4699:
4696:
4693:
4687:
4684:
4679:
4674:
4671:
4668:
4665:
4661:
4654:
4649:
4644:
4640:
4632:
4631:
4630:
4613:
4605:
4601:
4597:
4594:
4587:then for all
4574:
4571:
4568:
4561:moreover, if
4544:
4522:
4517:
4508:
4505:
4502:
4496:
4493:
4488:
4485:
4479:
4474:
4469:
4465:
4457:
4456:
4455:
4453:
4449:
4441:
4439:
4437:
4432:
4428:
4424:
4417:Log-concavity
4416:
4414:
4412:
4393:
4390:
4387:
4379:
4376:
4373:
4369:
4361:
4357:
4352:
4344:
4340:
4333:
4330:
4327:
4324:
4319:
4316:
4310:
4305:
4301:
4293:
4292:
4291:
4289:
4281:
4276:
4271:
4266:
4265:
4264:
4262:
4257:
4255:
4250:
4233:
4213:
4210:
4207:
4204:
4201:
4198:
4195:
4175:
4172:
4169:
4143:
4140:
4137:
4132:
4129:
4124:
4120:
4109:
4108:
4107:
4105:
4101:
4097:
4093:
4074:
4067:
4063:
4053:
4050:
4047:
4043:
4039:
4034:
4030:
4026:
4023:
4018:
4015:
4010:
4006:
4001:
3993:
3992:
3991:
3970:
3966:
3956:
3953:
3950:
3946:
3942:
3937:
3933:
3929:
3924:
3921:
3918:
3914:
3906:
3905:
3904:
3902:
3898:
3894:
3886:
3884:
3882:
3879:
3875:
3871:
3867:
3863:
3859:
3855:
3850:
3848:
3844:
3840:
3836:
3832:
3825:
3824:Taylor series
3806:
3800:
3797:
3791:
3787:
3774:
3771:
3768:
3764:
3758:
3755:
3750:
3745:
3741:
3733:
3732:
3731:
3730:
3722:
3720:
3703:
3697:
3692:
3688:
3684:
3678:
3671:
3668:
3660:
3654:
3578:
3575:
3513:
3492:
3489:
3451:
3450:
3449:
3447:
3443:
3439:
3435:
3431:
3426:
3408:
3403:
3400:
3395:
3391:
3386:
3382:
3377:
3373:
3366:
3363:
3357:
3353:
3340:
3337:
3334:
3330:
3326:
3320:
3314:
3307:
3306:
3305:
3303:
3295:
3293:
3291:
3273:
3270:
3267:
3259:
3256:
3248:
3245:
3242:
3238:
3233:
3228:
3225:
3220:
3214:
3209:
3206:
3203:
3199:
3195:
3190:
3187:
3184:
3180:
3174:
3170:
3158:
3155:
3142:
3137:
3134:
3131:
3127:
3118:
3104:
3101:
3098:
3078:
3075:
3072:
3063:
3048:
3045:
3042:
3038:
3032:
3029:
3026:
3018:
3015:
3012:
3004:
3000:
2988:
2985:
2972:
2967:
2964:
2961:
2957:
2953:
2948:
2944:
2938:
2935:
2932:
2928:
2922:
2918:
2906:
2903:
2890:
2885:
2882:
2879:
2875:
2866:
2852:
2849:
2846:
2837:
2824:
2819:
2816:
2813:
2809:
2803:
2800:
2797:
2789:
2786:
2783:
2780:
2772:
2768:
2756:
2753:
2740:
2735:
2732:
2729:
2725:
2719:
2716:
2713:
2705:
2702:
2695:
2690:
2687:
2682:
2676:
2671:
2668:
2665:
2661:
2657:
2652:
2648:
2642:
2639:
2636:
2632:
2626:
2622:
2610:
2607:
2594:
2589:
2586:
2583:
2579:
2570:
2568:
2563:
2550:
2545:
2542:
2539:
2536:
2533:
2529:
2523:
2520:
2517:
2509:
2506:
2492:
2489:
2476:
2471:
2468:
2465:
2461:
2457:
2452:
2449:
2446:
2442:
2430:
2427:
2414:
2409:
2406:
2403:
2399:
2390:
2387:
2374:
2369:
2366:
2363:
2359:
2353:
2350:
2347:
2339:
2336:
2322:
2319:
2306:
2301:
2298:
2295:
2291:
2287:
2282:
2278:
2269:
2267:
2248:
2243:
2239:
2233:
2230:
2227:
2223:
2211:
2208:
2196:
2191:
2188:
2183:
2177:
2172:
2169:
2166:
2162:
2156:
2151:
2148:
2145:
2141:
2137:
2132:
2129:
2126:
2122:
2114:
2113:
2112:
2110:
2106:
2104:
2099:
2096:into exactly
2095:
2078:
2073:
2070:
2065:
2041:
2037:
2032:
2029:
2024:
2018:
2013:
2010:
2007:
2003:
1999:
1994:
1990:
1982:
1981:
1980:
1979:
1974:
1972:
1965:
1940:
1937:
1920:
1916:
1912:
1908:
1889:
1884:
1880:
1868:
1865:
1852:
1847:
1844:
1841:
1837:
1833:
1828:
1825:
1822:
1818:
1810:
1809:
1808:
1806:
1802:
1794:
1789:
1787:
1784:
1767:
1762:
1757:
1752:
1747:
1740:
1735:
1730:
1725:
1718:
1713:
1708:
1701:
1696:
1689:
1677:
1657:
1654:
1651:
1647:
1638:
1634:
1625:
1621:
1606:
1603:
1600:
1597:
1594:
1586:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1545:
1541:
1536:
1533:
1530:
1527:
1524:
1520:
1511:
1508:
1505:
1501:
1488:
1485:
1481:
1463:
1460:
1457:
1454:
1451:
1447:
1438:
1435:
1432:
1428:
1419:
1404:
1401:
1396:
1393:
1390:
1386:
1377:
1376:
1375:
1373:
1369:
1365:
1361:
1357:
1356:Bell triangle
1348:
1343:
1342:Bell triangle
1335:
1333:
1331:
1326:
1322:
1318:
1314:
1309:
1307:
1303:
1299:
1292:
1285:
1281:
1277:
1273:
1269:
1261:
1259:
1257:
1253:
1249:
1245:
1244:rhyme schemes
1238:Rhyme schemes
1237:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1160:
1159:
1158:
1142:
1138:
1117:
1109:
1093:
1085:
1067:
1063:
1054:
1053:prime numbers
1038:
1030:
1026:
1010:
998:
996:
994:
990:
986:
982:
978:
974:
970:
965:
962:
958:
940:
936:
927:
923:
904:
895:
892:
889:
886:
883:
870:
856:
847:
844:
841:
835:
829:
816:
802:
793:
790:
787:
781:
775:
762:
748:
739:
736:
733:
727:
721:
708:
694:
685:
679:
673:
667:
661:
648:
647:
646:
629:
626:
623:
620:
617:
594:
591:
586:
582:
561:
541:
521:
501:
493:
475:
471:
457:
450:
446:
441:
436:
428:
423:
421:
419:
415:
399:
377:
373:
364:
360:
355:
354:-line poems.
341:
333:
332:rhyme schemes
315:
311:
302:
286:
264:
260:
247:
242:
225:
222:
219:
216:
213:
210:
207:
204:
201:
198:
195:
192:
189:
186:
183:
180:
177:
174:
171:
164:
163:
162:
148:
145:
140:
136:
132:
127:
123:
114:
110:
94:
72:
68:
58:
56:
52:
48:
44:
40:
33:
19:
9312:Transposable
9176:Narcissistic
9083:Digital root
9003:Super-Poulet
8963:Jordan–Pólya
8912:prime factor
8817:Noncototient
8784:Almost prime
8766:Superperfect
8741:Refactorable
8736:Quasiperfect
8711:Hyperperfect
8552:Pseudoprimes
8523:Wall–Sun–Sun
8458:Ordered Bell
8428:Fuss–Catalan
8392:
8340:non-centered
8290:Dodecahedral
8267:non-centered
8153:non-centered
8055:Wolstenholme
7800:× 2 ± 1
7797:
7796:Of the form
7763:Eighth power
7743:Fourth power
7633:
7576:
7570:
7566:
7535:
7496:
7490:
7458:
7454:
7414:
7408:
7382:(1): 15–57.
7379:
7373:
7353:
7349:
7318:
7305:
7272:
7261:
7242:(5): 24–30.
7239:
7233:
7216:
7186:(1): 67–78.
7183:
7182:. Series A.
7177:
7168:
7164:
7158:
7088:
7055:math/0011235
7045:
7041:
7014:
7013:. Series A.
7010:
6981:math/0507169
6971:
6967:
6954:
6942:
6938:
6922:
6918:
6895:
6868:
6864:
6835:
6831:
6802:
6798:
6775:
6769:
6729:math/0104137
6719:
6715:
6693:
6682:Gardner 1978
6677:
6661:
6649:
6637:
6610:
6597:
6586:. Retrieved
6573:
6561:
6549:
6537:
6525:
6514:. Retrieved
6507:the original
6498:
6491:
6479:
6467:
6455:
6443:
6431:
6404:
6347:
6343:
6318:
6306:
6294:
6281:
6269:
6257:
6249:
6241:
6213:
6206:
6199:Gardner 1978
6136:
6128:
6123:
6117:
6115:
6103:
6098:which gives
6085:
6077:
6058:
6054:
6053:). The next
6042:
6022:
6015:Gardner 1978
6013:
6001:
5708:
5529:
5099:
4730:
4560:
4445:
4435:
4430:
4426:
4420:
4408:
4285:
4260:
4258:
4161:
4103:
4099:
4095:
4091:
4089:
3989:
3901:prime number
3896:
3890:
3877:
3865:
3857:
3853:
3851:
3834:
3827:
3821:
3726:
3655:
3528:
3445:
3437:
3427:
3423:
3299:
3119:
3064:
2867:
2838:
2571:
2564:
2391:
2388:
2270:
2263:
2107:
2102:
2097:
2093:
2056:
1975:
1967:
1963:
1921:. There are
1918:
1914:
1910:
1906:
1904:
1798:
1785:
1678:
1675:
1623:
1483:
1479:
1363:
1359:
1353:
1320:
1316:
1313:permutations
1310:
1305:
1301:
1294:
1287:
1283:
1279:
1275:
1272:Gardner 1978
1265:
1262:Permutations
1247:
1241:
1106:. These are
1051:of distinct
1003:If a number
1002:
975:. These are
966:
919:
463:In general,
462:
448:
444:
356:
251:
59:
43:Bell numbers
42:
36:
18:Bell numbers
9333:Extravagant
9328:Equidigital
9283:permutation
9242:Palindromic
9216:Automorphic
9114:Sum-product
9093:Sum-product
9048:Persistence
8943:Erdős–Woods
8865:Untouchable
8746:Semiperfect
8696:Hemiperfect
8357:Tesseractic
8295:Icosahedral
8275:Tetrahedral
8206:Dodecagonal
7907:Recursively
7778:Prime power
7753:Sixth power
7748:Fifth power
7728:Power of 10
7686:Classes of
7579:: 323–327.
6861:Bell, E. T.
6828:Bell, E. T.
6698:Berndt 2011
6686:Berndt 2011
6151:Aitken 1933
6147:Peirce 1880
6023:Bell primes
6010:Bell primes
4442:Growth rate
2109:Spivey 2008
1624:Bell number
1486:-1)-th row)
32:Pell number
9545:Graphemics
9418:Pernicious
9272:Undulating
9247:Pandigital
9221:Trimorphic
8822:Nontotient
8671:Arithmetic
8285:Octahedral
8186:Heptagonal
8176:Pentagonal
8161:Triangular
8002:Sierpiński
7924:Jacobsthal
7723:Power of 3
7718:Power of 2
7558:0831.05001
7346:Moser, Leo
7338:0785.05001
7314:Lovász, L.
7171:: 333–336.
6945:(2): 8–13.
6707:References
6630:Knuth 2013
6588:2013-10-24
6530:Engel 1994
6516:2012-09-02
6055:Bell prime
4448:asymptotic
1803:involving
1790:Properties
1025:squarefree
989:transitive
492:partitions
238:(sequence
9302:Parasitic
9151:Factorion
9078:Digit sum
9070:Digit sum
8888:Fortunate
8875:Primorial
8789:Semiprime
8726:Practical
8691:Descartes
8686:Deficient
8676:Betrothed
8518:Wieferich
8347:Pentatope
8310:pyramidal
8201:Decagonal
8196:Nonagonal
8191:Octagonal
8181:Hexagonal
8040:Practical
7987:Congruent
7919:Fibonacci
7883:Loeschian
7635:MathWorld
7356:: 49–54.
7293:0906.0696
7279:EMS Press
7122:∑
6778:: 18–23.
6666:Rota 1964
6654:Bell 1938
6642:Bell 1934
6409:Wilf 1994
6397:Rota 1964
6366:0354-5180
6311:Wilf 1994
5983:∞
5980:→
5941:
5927:
5921:
5886:
5875:
5869:
5836:
5815:
5804:
5798:
5783:−
5777:
5771:
5765:−
5759:
5730:
5623:and each
5591:∞
5588:→
5553:
5490:−
5262:×
5235:π
5218:−
5193:
5184:×
5074:−
5068:−
5039:
4972:∼
4925:
4912:−
4889:
4883:
4877:−
4859:
4853:
4801:ε
4790:−
4755:ε
4688:
4675:ε
4666:−
4614:ε
4569:ε
4497:
4345:γ
4341:∫
4328:π
4173:≤
4141:−
4130:−
4024:≡
3930:≡
3881:cumulants
3780:∞
3765:∑
3576:≥
3490:≥
3401:−
3346:∞
3331:∑
3271:−
3257:−
3200:∑
3188:−
3128:∑
3065:and with
3030:−
3016:−
2958:∑
2936:−
2876:∑
2801:−
2784:−
2726:∑
2717:−
2703:−
2662:∑
2640:−
2580:∑
2569:include
2521:−
2507:−
2462:∑
2400:∑
2351:−
2337:−
2292:∑
2264:Applying
2231:−
2163:∑
2142:∑
2004:∑
1838:∑
1644:←
1565:−
1553:−
1534:−
1517:←
1455:−
1444:←
1268:shuffling
1219:×
1213:×
1201:×
1189:×
1177:×
1027:positive
985:symmetric
981:reflexive
979:that are
971:with its
957:empty set
226:…
9594:Category
9374:Friedman
9307:Primeval
9252:Repdigit
9209:-related
9156:Kaprekar
9130:Meertens
9053:Additive
9040:dynamics
8948:Friendly
8860:Sociable
8850:Amicable
8661:Abundant
8641:dynamics
8463:Schröder
8453:Narayana
8423:Eulerian
8413:Delannoy
8408:Dedekind
8229:centered
8095:centered
7982:Amenable
7939:Narayana
7929:Leonardo
7825:Mersenne
7773:Powerful
7713:Achilles
7533:(1994).
7485:(1996).
7281:. 2001 .
6953:(1981).
6764:(1933).
6754:16533831
6157:See also
6124:genji-ko
5973:as
5603:, where
4731:where
4446:Several
3672:′
424:Counting
87:, where
9547:related
9511:related
9475:related
9473:Sorting
9358:Vampire
9343:Harshad
9285:related
9257:Repunit
9171:Lychrel
9146:Dudeney
8998:Størmer
8993:Sphenic
8978:Regular
8916:divisor
8855:Perfect
8751:Sublime
8721:Perfect
8448:Motzkin
8403:Catalan
7944:Padovan
7878:Leyland
7873:Idoneal
7868:Hilbert
7840:Woodall
7601:0012612
7593:2305292
7523:1325876
7501:Bibcode
7475:2420912
7463:Bibcode
7439:0161805
7431:2312585
7396:2369442
7362:0078489
7244:Bibcode
7200:1255264
7072:1857258
7033:1354972
6998:2193154
6986:Bibcode
6885:1968633
6852:1968431
6819:2372336
6746:1831247
6605:(ed.).
6344:Filomat
6289:(ed.).
6252:(1909).
6233:0453532
6068:History
6049:in the
6046:A051130
6035:in the
6032:A051131
4273:in the
4270:A054767
4252:in the
4249:A001039
3899:is any
3872:of any
1362:or the
1055:, then
1029:integer
392:is the
359:moments
303:on it.
244:in the
241:A000110
109:integer
9413:Odious
9338:Frugal
9292:Cyclic
9281:Digit-
8988:Smooth
8973:Pronic
8933:Cyclic
8910:Other
8883:Euclid
8533:Wilson
8507:Primes
8166:Square
8035:Polite
7997:Riesel
7992:Knödel
7954:Perrin
7835:Thabit
7820:Fermat
7810:Cullen
7733:Square
7701:Powers
7599:
7591:
7556:
7546:
7521:
7473:
7437:
7429:
7394:
7360:
7336:
7326:
7198:
7100:
7070:
7031:
6996:
6906:
6883:
6850:
6817:
6752:
6744:
6364:
6231:
6221:
4832:
4812:
4739:
3870:moment
3839:moment
3529:Here,
1478:where
1366:after
1256:stanza
1250:-line
1246:of an
987:, and
107:is an
41:, the
9454:Prime
9449:Lucky
9438:sieve
9367:Other
9353:Smith
9233:Digit
9191:Happy
9166:Keith
9139:Other
8983:Rough
8953:Giuga
8418:Euler
8280:Cubic
7934:Lucas
7830:Proth
7649:(PDF)
7589:JSTOR
7540:(PDF)
7451:(PDF)
7427:JSTOR
7392:JSTOR
7288:arXiv
7224:–119.
7094:91–94
7050:arXiv
6976:arXiv
6935:(PDF)
6900:(PDF)
6881:JSTOR
6848:JSTOR
6815:JSTOR
6750:S2CID
6724:arXiv
6583:(PDF)
6510:(PDF)
6503:(PDF)
6184:Notes
4486:0.792
4226:, or
3903:then
3895:: If
3845:with
3841:of a
1023:is a
416:with
9408:Evil
9088:Self
9038:and
8928:Blum
8639:and
8443:Lobb
8398:Cake
8393:Bell
8143:Star
8050:Ulam
7949:Pell
7738:Cube
7544:ISBN
7324:ISBN
7157:für
7098:ISBN
6904:ISBN
6684:and
6611:The
6362:ISSN
6295:The
6219:ISBN
6149:and
6062:2841
6051:OEIS
6037:OEIS
5650:and
4824:and
4650:<
4598:>
4572:>
4475:<
4275:OEIS
4263:are
4254:OEIS
4188:and
3852:The
3434:urns
3300:The
1370:and
1252:poem
418:mean
334:for
246:OEIS
220:4140
113:zero
9526:Ban
8914:or
8433:Lah
7581:doi
7569:".
7554:Zbl
7509:doi
7419:doi
7384:doi
7334:Zbl
7252:doi
7240:238
7222:106
7188:doi
7060:doi
7019:doi
6873:doi
6840:doi
6807:doi
6780:doi
6734:doi
6352:doi
6137:In
6057:is
5577:as
5190:exp
5036:exp
4764:max
4669:0.6
4256:).
4234:173
4214:167
4208:163
4202:113
4176:101
4064:mod
3967:mod
3868:th
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