837:
2411:
2405:
3604:
126:
3664:
3652:
1596:
20:
6791:
4947:
35:
2395:
2689:
7100:
9034:
2063:
1414:
423:
1108:
4937:
2390:{\displaystyle T_{n}+T_{n-1}=\left({\frac {n^{2}}{2}}+{\frac {n}{2}}\right)+\left({\frac {\left(n-1\right)^{2}}{2}}+{\frac {n-1{\vphantom {\left(n-1\right)^{2}}}}{2}}\right)=\left({\frac {n^{2}}{2}}+{\frac {n}{2}}\right)+\left({\frac {n^{2}}{2}}-{\frac {n}{2}}\right)=n^{2}=(T_{n}-T_{n-1})^{2}.}
3599:
208:
590:, imagine a "half-rectangle" arrangement of objects corresponding to the triangular number, as in the figure below. Copying this arrangement and rotating it to create a rectangular figure doubles the number of objects, producing a rectangle with dimensions
4218:
5201:", describing them as "deeply intuitive" and "featured in an enormous number of games, incredibly versatile at providing escalating rewards for larger sets without overly incentivizing specialization to the exclusion of all other strategies".
2980:
3370:
3361:
2059:
Most simply, the sum of two consecutive triangular numbers is a square number, with the sum being the square of the difference between the two (and thus the difference of the two being the square root of the sum). Algebraically,
1409:{\displaystyle {\begin{aligned}\sum _{k=1}^{m}k+(m+1)&={\frac {m(m+1)}{2}}+m+1\\&={\frac {m(m+1)+2m+2}{2}}\\&={\frac {m^{2}+m+2m+2}{2}}\\&={\frac {m^{2}+3m+2}{2}}\\&={\frac {(m+1)(m+2)}{2}},\end{aligned}}}
4837:
1968:
4477:". This theorem does not imply that the triangular numbers are different (as in the case of 20 = 10 + 10 + 0), nor that a solution with exactly three nonzero triangular numbers must exist. This is a special case of the
969:
2047:
2817:
3794:
3221:
4735:
4653:
5013:. For example, a group stage with 4 teams requires 6 matches, and a group stage with 8 teams requires 28 matches. This is also equivalent to the handshake problem and fully connected network problems.
4295:
1075:
4095:
2503:
1113:
214:
3815:
The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus such numbers never end in 2, 4, 7, or 9. A final 3 must be preceded by a 0 or 5; a final 8 must be preceded by a 2 or 7.
4456:
418:{\displaystyle \displaystyle {\begin{aligned}T_{n}&=\sum _{k=1}^{n}k=1+2+\dotsb +n\\&={\frac {n^{2}+n{\vphantom {(n+1)}}}{2}}={\frac {n(n+1)}{2}}\\&={n+1 \choose 2}\end{aligned}}}
5527:
832:
1791:
474:
3651:
2618:
683:
2838:
6982:
4365:
7136:
4826:
3101:
769:
4571:
626:
104:
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666...
2684:
2536:
557:
3875:
The converse of the statement above is, however, not always true. For example, the digital root of 12, which is not a triangular number, is 3 and divisible by three.
2651:
6972:
4766:
1812:
628:, which is also the number of objects in the rectangle. Clearly, the triangular number itself is always exactly half of the number of objects in such a figure, or:
710:
588:
6625:
1462:
1101:
871:
3226:
1979:
1542:
1522:
1502:
1482:
1436:
991:
864:
514:
3826:
of a nonzero triangular number is always 1, 3, 6, or 9. Hence, every triangular number is either divisible by three or has a remainder of 1 when divided by 9:
2692:
A square whose side length is a triangular number can be partitioned into squares and half-squares whose areas add to cubes. This shows that the square of the
836:
7065:
2727:
1578:. However, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the
2427:
There are infinitely many triangular numbers that are also square numbers; e.g., 1, 36, 1225. Some of them can be generated by a simple recursive formula:
6906:
6260:
5743:
Ross, H.E. & Knott, B.I."Dicuil (9th century) on triangular and square numbers." British
Journal for the History of Mathematics, 2019,34 (2), 79-94.
4932:{\displaystyle {\begin{array}{ccccccc}&4&3&2&1&\\+&1&2&3&4&5\\\hline &5&5&5&5&5\end{array}}}
3692:
3812:
For example, the third triangular number is (3 × 2 =) 6, the seventh is (7 × 4 =) 28, the 31st is (31 × 16 =) 496, and the 127th is (127 × 64 =) 8128.
114:
3594:{\displaystyle \sum _{n_{k-1}=1}^{n_{k}}\sum _{n_{k-2}=1}^{n_{k-1}}\dots \sum _{n_{2}=1}^{n_{3}}\sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{k}+k-1}{k}}}
3029:(66) equals the fifth triangular number, 15. Every other triangular number is a hexagonal number. Knowing the triangular numbers, one can reckon any
7129:
5734:
Esposito, M. An unpublished astronomical treatise by the Irish monk Dicuil. Proceedings of the Royal Irish
Academy, XXXVI C. Dublin, 1907, 378-446.
5683:
6916:
4229:
9063:
3671:
6148:
6067:
5666:
5197:
designers
Geoffrey Engelstein and Isaac Shalev describe triangular numbers as having achieved "nearly the status of a mantra or koan among
4842:
6911:
2430:
7936:
7122:
3124:
4370:
7931:
7080:
6671:
6618:
1805:
The number of line segments between closest pairs of dots in the triangle can be represented in terms of the number of dots or with a
7946:
7926:
5478:
4659:
9078:
7060:
6962:
6952:
6032:
4577:
8639:
8219:
7070:
1707:
4461:
In 1796, Gauss discovered that every positive integer is representable as a sum of three triangular numbers (possibly including
996:
4478:
2549:
1678:
7941:
6253:
3872:
The digital root pattern for triangular numbers, repeating every nine terms, as shown above, is "1, 3, 6, 1, 6, 3, 1, 9, 9".
8725:
4493:
4302:
2410:
2404:
7075:
6977:
6611:
6184:
3948:, which yields all the odd squares are revealed by multiplying a triangular number by 8 and adding 1, and the process for
8041:
9073:
8391:
7710:
7503:
7103:
6438:
5880:
Liu, Zhi-Guo (2003-12-01). "An
Identity of Ramanujan and the Representation of Integers as Sums of Triangular Numbers".
8426:
8396:
8071:
8061:
3048:
8567:
7981:
7715:
7695:
7085:
6443:
6423:
6179:
8257:
8421:
6957:
6947:
6937:
5603:
9068:
8516:
8139:
7896:
7705:
7687:
7581:
7571:
7561:
6967:
6433:
6415:
6314:
6304:
6294:
4458:
both of which can easily be established either by looking at dot patterns (see above) or with some simple algebra.
8401:
2399:
This fact can be demonstrated graphically by positioning the triangles in opposite directions to create a square:
774:
9058:
8644:
8189:
7810:
7596:
7591:
7586:
7576:
7553:
6538:
6329:
6324:
6319:
6309:
6286:
6246:
5757:
5594:
th triangular number. However, although some other sources use this name and notation, they are not in wide use.
3030:
2709:
430:
9088:
7629:
5608:
4518:
4213:{\displaystyle \sum _{n=1}^{\infty }{1 \over {{n^{2}+n} \over 2}}=2\sum _{n=1}^{\infty }{1 \over {n^{2}+n}}=2.}
2543:
631:
7886:
6228:
8755:
8720:
8506:
8416:
8290:
8265:
8174:
8164:
7776:
7758:
7678:
7052:
6874:
6504:
6481:
6406:
4967:
4499:
9015:
8285:
8159:
7790:
7566:
7346:
7273:
6714:
6661:
6518:
6299:
4089:
3641:
843:
4507:
8270:
8124:
8051:
7206:
6921:
6666:
6573:
6098:
5691:
4503:
3627:, and its diagonals being simplex numbers; similarly, the fifth triangular number (15) equals the third
8979:
8619:
6174:
4778:
5611:, a triangular number whose position in the sequence of triangular numbers is also a triangular number
3960:
is an odd square is the inverse of this operation. The first several pairs of this form (not counting
717:
8912:
8806:
8770:
8511:
8234:
8214:
8031:
7700:
7488:
7460:
7032:
6869:
6638:
6428:
5803:
3624:
3364:
1548:
477:
130:
51:
3603:
1628:
people shakes hands once with each person. In other words, the solution to the handshake problem of
8634:
8498:
8493:
8461:
8224:
8199:
8194:
8169:
8099:
8095:
8026:
7916:
7748:
7544:
7513:
7012:
6879:
6471:
6277:
6102:
4530:
3674:
1973:
1806:
1599:
593:
560:
125:
9033:
1595:
9083:
9037:
8791:
8786:
8700:
8674:
8572:
8551:
8323:
8204:
8154:
8076:
8046:
7986:
7753:
7733:
7664:
7377:
6853:
6838:
6810:
6790:
6729:
6499:
6476:
6392:
6154:
6073:
5965:
5924:
5905:
5840:
4223:
3115:
2975:{\displaystyle \sum _{k=1}^{n}T_{k}=\sum _{k=1}^{n}{\frac {k(k+1)}{2}}={\frac {n(n+1)(n+2)}{6}}.}
2832:
157:
7921:
6942:
5713:
5105:
the current year in the depreciation schedule. Under this method, an item with a usable life of
3663:
6191:
19:
8931:
8876:
8730:
8705:
8679:
8456:
8134:
8129:
8056:
8036:
8021:
7743:
7725:
7644:
7634:
7397:
7382:
7042:
6843:
6815:
6759:
6739:
6724:
6466:
6453:
6372:
6362:
6211:
6144:
6063:
6028:
5897:
5832:
5662:
5532:
3022:
2656:
2508:
519:
5656:
5634:
2623:
8967:
8760:
8346:
8318:
8308:
8300:
8184:
8149:
8144:
8111:
7805:
7768:
7659:
7654:
7649:
7639:
7611:
7498:
7450:
7445:
7402:
7341:
7027:
6848:
6774:
6764:
6744:
6646:
6533:
6491:
6387:
6382:
6377:
6367:
6344:
6136:
6055:
6020:
5957:
5889:
5824:
4741:
3628:
3026:
2991:
166:
94:
4946:
4513:
Formulas involving expressing an integer as the sum of triangular numbers are connected to
688:
566:
8943:
8832:
8765:
8691:
8614:
8588:
8406:
8119:
7976:
7911:
7881:
7871:
7866:
7532:
7440:
7387:
7231:
7171:
6805:
6734:
6269:
6200:
5923:
Sun, Zhi-Hong (2016-01-24). "Ramanujan's theta functions and sums of triangular numbers".
5868:
5857:
4988:
cables or other connections; this is equivalent to the handshake problem mentioned above.
4772:
3356:{\displaystyle \sum _{n_{2}=1}^{n_{3}}\sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{3}+2}{3}},}
55:
5893:
1441:
1080:
1602:
that the number of possible handshakes between n people is the (n−1)th triangular number
34:
8948:
8816:
8801:
8665:
8629:
8604:
8480:
8451:
8436:
8313:
8209:
8179:
7906:
7861:
7738:
7336:
7331:
7326:
7298:
7283:
7196:
7181:
7159:
7146:
7037:
7022:
7017:
6696:
6681:
6461:
6094:
5652:
5540:
4514:
3806:
3686:
1527:
1507:
1487:
1467:
1421:
976:
849:
499:
139:
84:
63:
6214:
5991:
2688:
9052:
8871:
8855:
8796:
8750:
8446:
8431:
8341:
8066:
7624:
7493:
7455:
7412:
7293:
7278:
7268:
7226:
7216:
7191:
7002:
6676:
6583:
6357:
6077:
5909:
5844:
4489:
4471:
3809:. No odd perfect numbers are known; hence, all known perfect numbers are triangular.
2421:
59:
6158:
8907:
8896:
8811:
8649:
8624:
8541:
8441:
8411:
8386:
8370:
8275:
8242:
7991:
7965:
7876:
7815:
7392:
7288:
7221:
7201:
7176:
7007:
6749:
6691:
6588:
6543:
6204:
6195:
5869:
Fang: Nonexistence of a geometric progression that contains four triangular numbers
5575:
5021:
5017:
3823:
1674:
1579:
4962:-th triangular number plus one, forming the lazy caterer's sequence (OEIS A000124)
2416:
The double of a triangular number, as in the visual proof from the above section
8866:
8741:
8546:
8010:
7901:
7856:
7851:
7601:
7508:
7407:
7236:
7211:
7186:
6754:
6701:
6578:
6334:
5744:
5443:
5198:
4992:
5828:
3682:
Alternating triangular numbers (1, 6, 15, 28, ...) are also hexagonal numbers.
2056:
Triangular numbers have a wide variety of relations to other figurate numbers.
1963:{\displaystyle L_{n}=3T_{n-1}=3{n \choose 2};~~~L_{n}=L_{n-1}+3(n-1),~L_{1}=0.}
9003:
8984:
8280:
7891:
6140:
5194:
4502:
posed the question as to the existence of four distinct triangular numbers in
72:
th triangular number is the number of dots in the triangular arrangement with
7114:
5901:
5836:
5041:
is the length in years of the asset's useful life. Each year, the item loses
1551:, is said to have found this relationship in his early youth, by multiplying
8609:
8536:
8528:
8333:
8247:
7365:
6686:
6558:
6219:
5614:
5579:
4829:
1653:
964:{\displaystyle T_{1}=\sum _{k=1}^{1}k={\frac {1(1+1)}{2}}={\frac {2}{2}}=1.}
195:
185:
175:
98:
6049:
6059:
2042:{\displaystyle \lim _{n\to \infty }{\frac {T_{n}}{L_{n}}}={\frac {1}{3}}.}
8710:
6634:
5617:, an arrangement of ten points in a triangle, important in Pythagoreanism
1587:
1582:
in the 5th century BC. The two formulas were described by the Irish monk
4524:
The sum of two consecutive triangular numbers is a square number since:
3607:
The fourth triangular number equals the third tetrahedral number as the
8715:
8374:
6603:
6024:
5969:
3819:
2812:{\displaystyle \sum _{k=1}^{n}k^{3}=\left(\sum _{k=1}^{n}k\right)^{2}.}
480:. It represents the number of distinct pairs that can be selected from
5765:
6016:
Algebra in
Context: Introductory Algebra from Origins to Applications
6014:
5206:
1583:
203:
The triangular numbers are given by the following explicit formulas:
5961:
6238:
5929:
5758:"The Handshake Problem | National Association of Math Circles"
5661:(4th ed.). Houston, Texas: Publish or Perish. pp. 21–22.
1621:
of counting the number of handshakes if each person in a room with
6107:, vol. 1 (2nd ed.), J. Johnson and Co., pp. 332–335
5945:
4945:
4775:, is visually demonstrated in the following sum, which represents
3789:{\displaystyle M_{p}2^{p-1}={\frac {M_{p}(M_{p}+1)}{2}}=T_{M_{p}}}
3602:
3216:{\displaystyle \sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{2}+1}{2}}}
2687:
1594:
33:
18:
6119:
The Art of
Computer Programming: Volume 1: Fundamental Algorithms
4510:
to be impossible and was later proven by Fang and Chen in 2007.
9001:
8965:
8929:
8893:
8853:
8478:
8367:
8093:
8008:
7963:
7840:
7530:
7477:
7429:
7363:
7315:
7253:
7157:
7118:
6607:
6242:
5205:
Relationship between the maximum number of pips on an end of a
4730:{\displaystyle ={\frac {1}{2}}\,n{\Bigl (}(n-1)+(n+1){\Bigr )}}
1976:, the ratio between the two numbers, dots and line segments is
4648:{\displaystyle ={\frac {1}{2}}\,n(n-1)+{\frac {1}{2}}\,n(n+1)}
5714:"Webpage cites AN INTRODUCTION TO THE HISTORY OF MATHEMATICS"
5550:
is a square. Equivalently, if the positive triangular root
3689:
is triangular (as well as hexagonal), given by the formula
3677:
that all hexagonal numbers are odd-sided triangular numbers
1590:. An English translation of Dicuil's account is available.
109:
4290:{\displaystyle \sum _{n=1}^{\infty }{1 \over {n(n+1)}}=1.}
3640:
Triangular numbers correspond to the first-degree case of
1070:{\displaystyle T_{m}=\sum _{k=1}^{m}k={\frac {m(m+1)}{2}}}
2718:
is the same as the sum of the cubes of the integers 1 to
6235:, some higher dimensions, and some approximate formulas
5858:
Chen, Fang: Triangular numbers in geometric progression
4995:, the number of matches that need to be played between
3114:
The positive difference of two triangular numbers is a
1571:
pairs of numbers in the sum by the values of each pair
129:
Derivation of triangular numbers from a left-justified
5087:
is the item's beginning value (in units of currency),
2698:
th triangular number is equal to the sum of the first
434:
6983:
1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
6973:
1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
5481:
5099:
is the total number of years the item is usable, and
4840:
4781:
4744:
4662:
4580:
4533:
4373:
4305:
4232:
4098:
3695:
3373:
3229:
3127:
3051:
2841:
2730:
2659:
2626:
2552:
2511:
2433:
2066:
1982:
1815:
1710:
1530:
1510:
1490:
1470:
1444:
1424:
1111:
1083:
999:
979:
874:
852:
777:
720:
691:
634:
596:
569:
522:
502:
433:
212:
211:
6013:
Shell-Gellasch, Amy; Thoo, John (October 15, 2015).
5175:
in the fourth, accumulating a total depreciation of
4771:
This property, colloquially known as the theorem of
4299:
Two other formulas regarding triangular numbers are
23:
The first six triangular numbers (not starting with
8825:
8779:
8739:
8690:
8664:
8597:
8581:
8560:
8527:
8492:
8332:
8299:
8256:
8233:
8110:
7798:
7789:
7767:
7724:
7686:
7677:
7610:
7552:
7543:
7051:
6995:
6930:
6899:
6892:
6862:
6831:
6824:
6798:
6710:
6654:
6645:
6566:
6556:
6526:
6517:
6490:
6452:
6414:
6405:
6343:
6285:
6276:
6201:
There exist triangular numbers that are also square
5450:, one can define the (positive) triangular root of
2498:{\displaystyle S_{n+1}=4S_{n}\left(8S_{n}+1\right)}
2408: 10 + 15 = 25
6048:Engelstein, Geoffrey; Shalev, Isaac (2019-06-25).
5521:
4931:
4820:
4760:
4729:
4647:
4565:
4450:
4359:
4289:
4212:
3788:
3593:
3355:
3215:
3095:
2974:
2811:
2678:
2645:
2612:
2530:
2497:
2389:
2041:
1962:
1785:
1536:
1516:
1496:
1476:
1456:
1430:
1408:
1095:
1069:
985:
963:
858:
826:
763:
704:
677:
620:
582:
551:
508:
468:
417:
78:dots on each side, and is equal to the sum of the
5438:Triangular roots and tests for triangular numbers
4722:
4682:
4451:{\displaystyle T_{ab}=T_{a}T_{b}+T_{a-1}T_{b-1},}
3585:
3551:
3367:, can be generalized. This leads to the formula:
3344:
3316:
3207:
3179:
1870:
1857:
404:
383:
6231:by Rob Hubbard, including the generalisation to
1984:
6121:. 3rd Ed. Addison Wesley Longman, U.S.A. p. 48.
6019:. Johns Hopkins University Press. p. 210.
5522:{\displaystyle n={\frac {{\sqrt {8x+1}}-1}{2}}}
4991:In a tournament format that uses a round-robin
4470:= 0), writing in his diary his famous words, "
4222:This can be shown by using the basic sum of a
7130:
6619:
6254:
6089:
6087:
5745:https://doi.org/10.1080/26375451.2019.1598687
4506:. It was conjectured by Polish mathematician
3021:th triangular number. For example, the sixth
459:
438:
8:
7066:Hypergeometric function of a matrix argument
3931:will always be a triangular number, because
827:{\displaystyle T_{4}={\frac {4(4+1)}{2}}=10}
6922:1 + 1/2 + 1/3 + ... (Riemann zeta function)
5574:As stated, an alternative name proposed by
4976:computing devices requires the presence of
2984:More generally, the difference between the
1786:{\displaystyle 10?=1+2+3+4+5+6+7+8+9+10=55}
8998:
8962:
8926:
8890:
8850:
8524:
8489:
8475:
8364:
8107:
8090:
8005:
7960:
7837:
7795:
7683:
7549:
7540:
7527:
7474:
7431:Possessing a specific set of other numbers
7426:
7360:
7312:
7250:
7154:
7137:
7123:
7115:
6896:
6828:
6651:
6626:
6612:
6604:
6563:
6523:
6411:
6282:
6261:
6247:
6239:
5127:of its "losable" value in the first year,
4484:The largest triangular number of the form
3662:
842:This formula can be proven formally using
469:{\displaystyle \textstyle {n+1 \choose 2}}
6978:1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
5928:
5791:. Vol. 1 (3rd ed.). p. 48.
5491:
5488:
5480:
4841:
4839:
4812:
4799:
4786:
4780:
4752:
4743:
4721:
4720:
4681:
4680:
4676:
4666:
4661:
4626:
4616:
4594:
4584:
4579:
4557:
4538:
4532:
4433:
4417:
4404:
4394:
4378:
4372:
4342:
4329:
4310:
4304:
4259:
4254:
4248:
4237:
4231:
4189:
4184:
4179:
4173:
4162:
4132:
4127:
4125:
4120:
4114:
4103:
4097:
4092:of all the nonzero triangular numbers is
3778:
3773:
3745:
3732:
3725:
3710:
3700:
3694:
3584:
3561:
3550:
3548:
3539:
3527:
3522:
3509:
3504:
3492:
3487:
3474:
3469:
3448:
3443:
3424:
3419:
3407:
3402:
3383:
3378:
3372:
3343:
3326:
3315:
3313:
3304:
3292:
3287:
3274:
3269:
3257:
3252:
3239:
3234:
3228:
3206:
3189:
3178:
3176:
3167:
3155:
3150:
3137:
3132:
3126:
3121:The pattern found for triangular numbers
3075:
3059:
3050:
3045:-gonal number is obtained by the formula
2927:
2897:
2891:
2880:
2867:
2857:
2846:
2840:
2800:
2786:
2775:
2756:
2746:
2735:
2729:
2664:
2658:
2631:
2625:
2613:{\displaystyle S_{n}=34S_{n-1}-S_{n-2}+2}
2592:
2573:
2557:
2551:
2516:
2510:
2478:
2460:
2438:
2432:
2378:
2362:
2349:
2333:
2311:
2297:
2291:
2268:
2254:
2248:
2219:
2196:
2195:
2183:
2169:
2147:
2124:
2110:
2104:
2084:
2071:
2065:
2026:
2015:
2005:
1999:
1987:
1981:
1948:
1905:
1892:
1869:
1856:
1854:
1836:
1820:
1814:
1709:
1529:
1509:
1489:
1469:
1443:
1423:
1360:
1323:
1316:
1273:
1266:
1214:
1165:
1131:
1120:
1112:
1110:
1082:
1040:
1028:
1017:
1004:
998:
978:
973:Now assume that, for some natural number
945:
915:
903:
892:
879:
873:
851:
791:
782:
776:
728:
719:
696:
690:
678:{\displaystyle T_{n}={\frac {n(n+1)}{2}}}
648:
639:
633:
595:
574:
568:
541:
521:
501:
458:
437:
435:
432:
403:
382:
380:
343:
312:
311:
299:
292:
249:
238:
221:
213:
210:
97:of triangular numbers, starting with the
5946:"Triangular Numbers and Perfect Squares"
5802:Baumann, Michael Heinrich (2018-12-12).
5203:
5001:teams is equal to the triangular number
1547:The German mathematician and scientist,
124:
6051:Building Blocks of Tabletop Game Design
5626:
3623:-simplex number due to the symmetry of
4360:{\displaystyle T_{a+b}=T_{a}+T_{b}+ab}
1669:This same function was coined as the "
6133:Algorithms for Functional Programming
5209:and the number of dominoes in its set
1524:, and ultimately all natural numbers
7:
5944:Beldon, Tom; Gardiner, Tony (2002).
1795:which of course, corresponds to the
6943:1 − 1 + 1 − 1 + ⋯ (Grandi's series)
5690:. Computing Science. Archived from
5531:which follows immediately from the
3894:is also a triangular number, given
2052:Relations to other figurate numbers
1687:(analog for the factorial notation
54:. Triangular numbers are a type of
5894:10.1023/B:RAMA.0000012425.42327.ae
5582:, is "termial", with the notation
5191:(the whole) of the losable value.
4249:
4174:
4115:
3555:
3320:
3183:
1994:
1861:
487:objects, and it is read aloud as "
442:
387:
14:
7061:Generalized hypergeometric series
5810:-dimensionale Champagnerpyramide"
4821:{\displaystyle T_{4}+T_{5}=5^{2}}
3096:{\displaystyle Ck_{n}=kT_{n-1}+1}
771:(green plus yellow) implies that
9032:
8640:Perfect digit-to-digit invariant
7099:
7098:
7071:Lauricella hypergeometric series
6789:
6229:Hypertetrahedral Polytopic Roots
3650:
2724:. This can also be expressed as
2409:
2403:
835:
764:{\displaystyle 2T_{4}=4(4+1)=20}
7081:Riemann's differential equation
5789:The Art of Computer Programming
4479:Fermat polygonal number theorem
2402:6 + 10 = 16
1679:The Art of Computer Programming
1464:. Since it is clearly true for
5817:Mathematische Semesterberichte
4950:The maximum number of pieces,
4717:
4705:
4699:
4687:
4642:
4630:
4610:
4598:
4275:
4263:
3757:
3738:
2960:
2948:
2945:
2933:
2915:
2903:
2375:
2342:
1991:
1935:
1923:
1652:is the additive analog of the
1418:so if the formula is true for
1390:
1378:
1375:
1363:
1232:
1220:
1183:
1171:
1155:
1143:
1058:
1046:
933:
921:
809:
797:
752:
740:
666:
654:
615:
603:
563:. For every triangular number
538:
526:
361:
349:
327:
315:
50:counts objects arranged in an
1:
9064:Factorial and binomial topics
7479:Expressible via specific sums
7076:Modular hypergeometric series
6917:1/4 + 1/16 + 1/64 + 1/256 + ⋯
6439:Centered dodecahedral numbers
4566:{\displaystyle T_{n-1}+T_{n}}
3884:is a triangular number, then
2546:are found from the recursion
621:{\displaystyle n\times (n+1)}
6444:Centered icosahedral numbers
6424:Centered tetrahedral numbers
5635:"Triangular Number Sequence"
5211:(values in bold are common)
5093:is its final salvage value,
4500:Wacław Franciszek Sierpiński
3223:and for tetrahedral numbers
2417:
2197:
516:th triangular number equals
313:
8568:Multiplicative digital root
7086:Theta hypergeometric series
6434:Centered octahedral numbers
6315:Centered heptagonal numbers
6305:Centered pentagonal numbers
6295:Centered triangular numbers
6180:Encyclopedia of Mathematics
5022:sum-of-years' digits method
5016:One way of calculating the
3615:-simplex number equals the
1660:of integers from 1 to
1484:, it is therefore true for
834:(green).
559:can be illustrated using a
9105:
6968:Infinite arithmetic series
6912:1/2 + 1/4 + 1/8 + 1/16 + ⋯
6907:1/2 − 1/4 + 1/8 − 1/16 + ⋯
6539:Squared triangular numbers
6330:Centered decagonal numbers
6325:Centered nonagonal numbers
6320:Centered octagonal numbers
6310:Centered hexagonal numbers
6131:Stone, John David (2018),
5829:10.1007/s00591-018-00236-x
5684:"Gauss's Day of Reckoning"
2825:triangular numbers is the
9028:
9011:
8997:
8975:
8961:
8939:
8925:
8903:
8889:
8862:
8849:
8645:Perfect digital invariant
8488:
8474:
8382:
8363:
8220:Superior highly composite
8106:
8089:
8017:
8004:
7972:
7959:
7847:
7836:
7539:
7526:
7484:
7473:
7436:
7425:
7373:
7359:
7322:
7311:
7264:
7249:
7167:
7153:
7094:
6787:
6141:10.1007/978-3-662-57970-1
6135:, Springer, p. 282,
6117:Donald E. Knuth (1997).
5024:, which involves finding
4494:Ramanujan–Nagell equation
3031:centered polygonal number
2544:square triangular numbers
846:. It is clearly true for
9079:Squares in number theory
8258:Euler's totient function
8042:Euler–Jacobi pseudoprime
7317:Other polynomial numbers
6505:Square pyramidal numbers
6482:Stella octangula numbers
5950:The Mathematical Gazette
5609:Doubly triangular number
4519:Ramanujan theta function
3111:is a triangular number.
2679:{\displaystyle S_{1}=1.}
2531:{\displaystyle S_{1}=1.}
552:{\displaystyle n(n+1)/2}
8072:Somer–Lucas pseudoprime
8062:Lucas–Carmichael number
7897:Lazy caterer's sequence
6799:Properties of sequences
6300:Centered square numbers
4968:fully connected network
4033:, these formulas yield
2646:{\displaystyle S_{0}=0}
1798:tenth triangular number
1656:function, which is the
58:, other examples being
38:Triangular Numbers Plot
7947:Wedderburn–Etherington
7347:Lucky numbers of Euler
6662:Arithmetic progression
6099:Lagrange, Joseph Louis
6003:See equations 18 - 20.
5566:th triangular number.
5523:
4963:
4933:
4822:
4762:
4761:{\displaystyle =n^{2}}
4731:
4649:
4567:
4452:
4361:
4291:
4253:
4214:
4178:
4119:
3790:
3632:
3595:
3534:
3499:
3461:
3414:
3357:
3299:
3264:
3217:
3162:
3097:
2976:
2896:
2862:
2813:
2791:
2751:
2705:
2680:
2647:
2614:
2532:
2499:
2391:
2043:
1964:
1787:
1606:The triangular number
1603:
1538:
1518:
1498:
1478:
1458:
1432:
1410:
1136:
1097:
1071:
1033:
987:
965:
908:
860:
844:mathematical induction
828:
765:
706:
679:
622:
584:
553:
510:
493:plus one choose two".
470:
419:
254:
200:
39:
31:
8235:Prime omega functions
8052:Frobenius pseudoprime
7842:Combinatorial numbers
7711:Centered dodecahedral
7504:Primary pseudoperfect
7053:Hypergeometric series
6667:Geometric progression
6429:Centered cube numbers
6233:triangular cube roots
6060:10.1201/9780429430701
5882:The Ramanujan Journal
5524:
5111:= 4 years would lose
4958:straight cuts is the
4949:
4934:
4823:
4763:
4732:
4650:
4568:
4504:geometric progression
4453:
4362:
4292:
4233:
4215:
4158:
4099:
3900:is an odd square and
3791:
3606:
3596:
3500:
3465:
3415:
3374:
3365:binomial coefficients
3358:
3265:
3230:
3218:
3128:
3098:
3025:(81) minus the sixth
3013:-gonal number is the
2977:
2876:
2842:
2821:The sum of the first
2814:
2771:
2731:
2691:
2681:
2648:
2615:
2533:
2500:
2392:
2044:
1965:
1788:
1598:
1539:
1519:
1499:
1479:
1459:
1433:
1411:
1116:
1098:
1072:
1013:
988:
966:
888:
861:
829:
766:
707:
705:{\displaystyle T_{4}}
680:
623:
585:
583:{\displaystyle T_{n}}
554:
511:
471:
420:
234:
128:
99:0th triangular number
37:
22:
8694:-composition related
8494:Arithmetic functions
8096:Arithmetic functions
8032:Elliptic pseudoprime
7716:Centered icosahedral
7696:Centered tetrahedral
7033:Trigonometric series
6825:Properties of series
6672:Harmonic progression
6472:Dodecahedral numbers
5558:is an integer, then
5479:
5442:By analogy with the
4838:
4779:
4742:
4660:
4578:
4531:
4517:, in particular the
4371:
4303:
4230:
4096:
3693:
3371:
3227:
3125:
3049:
2839:
2728:
2716:th triangular number
2657:
2624:
2550:
2509:
2431:
2225:
2064:
1980:
1813:
1708:
1586:in about 816 in his
1549:Carl Friedrich Gauss
1528:
1508:
1488:
1468:
1442:
1422:
1109:
1081:
997:
977:
872:
850:
775:
718:
689:
632:
594:
567:
520:
500:
478:binomial coefficient
431:
330:
209:
52:equilateral triangle
9074:Proof without words
8620:Kaprekar's constant
8140:Colossally abundant
8027:Catalan pseudoprime
7927:Schröder–Hipparchus
7706:Centered octahedral
7582:Centered heptagonal
7572:Centered pentagonal
7562:Centered triangular
7162:and related numbers
7013:Formal power series
6589:8-hypercube numbers
6584:7-hypercube numbers
6579:6-hypercube numbers
6574:5-hypercube numbers
6544:Tesseractic numbers
6500:Tetrahedral numbers
6477:Icosahedral numbers
6393:Dodecagonal numbers
6215:"Triangular Number"
6175:"Arithmetic series"
6104:Elements of Algebra
5994:. Wolfram MathWorld
5992:"Triangular Number"
5990:Eric W. Weisstein.
5212:
5020:of an asset is the
4508:Kazimierz Szymiczek
3675:Proof without words
3642:Faulhaber's formula
2226:
2198:
1807:recurrence relation
1600:Proof without words
1457:{\displaystyle m+1}
1096:{\displaystyle m+1}
331:
314:
158:Tetrahedral numbers
9038:Mathematics portal
8980:Aronson's sequence
8726:Smarandache–Wellin
8483:-dependent numbers
8190:Primitive abundant
8077:Strong pseudoprime
8067:Perrin pseudoprime
8047:Fermat pseudoprime
7987:Wolstenholme prime
7811:Squared triangular
7597:Centered decagonal
7592:Centered nonagonal
7587:Centered octagonal
7577:Centered hexagonal
6811:Monotonic function
6730:Fibonacci sequence
6467:Octahedral numbers
6373:Heptagonal numbers
6363:Pentagonal numbers
6353:Triangular numbers
6212:Weisstein, Eric W.
6192:Triangular numbers
6025:10.1353/book.49475
5688:American Scientist
5519:
5204:
5159:in the third, and
4964:
4929:
4927:
4818:
4758:
4727:
4645:
4563:
4448:
4357:
4287:
4224:telescoping series
4210:
3786:
3658:{{{annotations}}}
3633:
3591:
3353:
3213:
3116:trapezoidal number
3093:
2972:
2833:tetrahedral number
2809:
2706:
2676:
2643:
2610:
2528:
2495:
2387:
2039:
1998:
1960:
1783:
1702:is equivalent to:
1604:
1534:
1514:
1494:
1474:
1454:
1428:
1406:
1404:
1093:
1067:
983:
961:
856:
824:
761:
702:
675:
618:
580:
549:
506:
496:The fact that the
476:is notation for a
466:
465:
415:
414:
412:
201:
149:Triangular numbers
40:
32:
9069:Integer sequences
9046:
9045:
9024:
9023:
8993:
8992:
8957:
8956:
8921:
8920:
8885:
8884:
8845:
8844:
8841:
8840:
8660:
8659:
8470:
8469:
8359:
8358:
8355:
8354:
8301:Aliquot sequences
8112:Divisor functions
8085:
8084:
8057:Lucas pseudoprime
8037:Euler pseudoprime
8022:Carmichael number
8000:
7999:
7955:
7954:
7832:
7831:
7828:
7827:
7824:
7823:
7785:
7784:
7673:
7672:
7630:Square triangular
7522:
7521:
7469:
7468:
7421:
7420:
7355:
7354:
7307:
7306:
7245:
7244:
7112:
7111:
7043:Generating series
6991:
6990:
6963:1 − 2 + 4 − 8 + ⋯
6958:1 + 2 + 4 + 8 + ⋯
6953:1 − 2 + 3 − 4 + ⋯
6948:1 + 2 + 3 + 4 + ⋯
6938:1 + 1 + 1 + 1 + ⋯
6888:
6887:
6816:Periodic sequence
6785:
6784:
6770:Triangular number
6760:Pentagonal number
6740:Heptagonal number
6725:Complete sequence
6647:Integer sequences
6601:
6600:
6597:
6596:
6552:
6551:
6534:Pentatope numbers
6513:
6512:
6401:
6400:
6388:Decagonal numbers
6383:Nonagonal numbers
6378:Octagonal numbers
6368:Hexagonal numbers
6150:978-3-662-57968-8
6069:978-0-429-43070-1
5668:978-0-914098-91-1
5604:1 + 2 + 3 + 4 + ⋯
5533:quadratic formula
5517:
5505:
5435:
5434:
4674:
4624:
4592:
4279:
4202:
4150:
4148:
4016:, ... etc. Given
3764:
3625:Pascal's triangle
3583:
3342:
3205:
3023:heptagonal number
2967:
2922:
2319:
2306:
2276:
2263:
2233:
2178:
2132:
2119:
2034:
2021:
1983:
1943:
1887:
1884:
1881:
1868:
1619:handshake problem
1537:{\displaystyle n}
1517:{\displaystyle 3}
1497:{\displaystyle 2}
1477:{\displaystyle 1}
1438:, it is true for
1431:{\displaystyle m}
1397:
1348:
1304:
1254:
1190:
1065:
986:{\displaystyle m}
953:
940:
859:{\displaystyle 1}
816:
673:
509:{\displaystyle n}
457:
402:
368:
338:
167:Pentatope numbers
131:Pascal's triangle
44:triangular number
9096:
9059:Figurate numbers
9036:
8999:
8968:Natural language
8963:
8927:
8895:Generated via a
8891:
8851:
8756:Digit-reassembly
8721:Self-descriptive
8525:
8490:
8476:
8427:Lucas–Carmichael
8417:Harmonic divisor
8365:
8291:Sparsely totient
8266:Highly cototient
8175:Multiply perfect
8165:Highly composite
8108:
8091:
8006:
7961:
7942:Telephone number
7838:
7796:
7777:Square pyramidal
7759:Stella octangula
7684:
7550:
7541:
7533:Figurate numbers
7528:
7475:
7427:
7361:
7313:
7251:
7155:
7139:
7132:
7125:
7116:
7102:
7101:
7028:Dirichlet series
6897:
6829:
6793:
6765:Polygonal number
6745:Hexagonal number
6718:
6652:
6628:
6621:
6614:
6605:
6564:
6524:
6412:
6283:
6270:Figurate numbers
6263:
6256:
6249:
6240:
6225:
6224:
6188:
6162:
6161:
6128:
6122:
6115:
6109:
6108:
6091:
6082:
6081:
6045:
6039:
6038:
6010:
6004:
6002:
6000:
5999:
5987:
5981:
5980:
5978:
5976:
5956:(507): 423–431.
5941:
5935:
5934:
5932:
5920:
5914:
5913:
5877:
5871:
5866:
5860:
5855:
5849:
5848:
5814:
5809:
5799:
5793:
5792:
5784:
5778:
5777:
5775:
5773:
5768:on 10 March 2016
5764:. Archived from
5754:
5748:
5741:
5735:
5732:
5726:
5725:
5723:
5721:
5709:
5703:
5702:
5700:
5699:
5679:
5673:
5672:
5649:
5643:
5642:
5631:
5593:
5587:
5578:, by analogy to
5570:Alternative name
5565:
5561:
5557:
5553:
5549:
5538:
5535:. So an integer
5528:
5526:
5525:
5520:
5518:
5513:
5506:
5492:
5489:
5474:
5459:
5453:
5449:
5213:
5190:
5188:
5187:
5184:
5181:
5174:
5172:
5171:
5168:
5165:
5158:
5156:
5155:
5152:
5149:
5142:
5140:
5139:
5136:
5133:
5126:
5124:
5123:
5120:
5117:
5110:
5104:
5098:
5092:
5086:
5080:
5079:
5077:
5076:
5066:
5063:
5040:
5034:
5012:
5000:
4987:
4975:
4954:obtainable with
4938:
4936:
4935:
4930:
4928:
4900:
4865:
4844:
4827:
4825:
4824:
4819:
4817:
4816:
4804:
4803:
4791:
4790:
4767:
4765:
4764:
4759:
4757:
4756:
4736:
4734:
4733:
4728:
4726:
4725:
4686:
4685:
4675:
4667:
4654:
4652:
4651:
4646:
4625:
4617:
4593:
4585:
4572:
4570:
4569:
4564:
4562:
4561:
4549:
4548:
4487:
4476:
4469:
4457:
4455:
4454:
4449:
4444:
4443:
4428:
4427:
4409:
4408:
4399:
4398:
4386:
4385:
4366:
4364:
4363:
4358:
4347:
4346:
4334:
4333:
4321:
4320:
4296:
4294:
4293:
4288:
4280:
4278:
4255:
4252:
4247:
4219:
4217:
4216:
4211:
4203:
4201:
4194:
4193:
4180:
4177:
4172:
4151:
4149:
4144:
4137:
4136:
4126:
4121:
4118:
4113:
4084:
4071:
4058:
4045:
4032:
4021:
4015:
4007:
3999:
3991:
3983:
3975:
3967:
3959:
3953:
3947:
3930:
3924:
3923:
3921:
3920:
3917:
3914:
3899:
3893:
3883:
3867:91 = 9 × 10 + 1
3804:
3795:
3793:
3792:
3787:
3785:
3784:
3783:
3782:
3765:
3760:
3750:
3749:
3737:
3736:
3726:
3721:
3720:
3705:
3704:
3666:
3654:
3636:Other properties
3629:pentatope number
3600:
3598:
3597:
3592:
3590:
3589:
3588:
3579:
3566:
3565:
3554:
3544:
3543:
3533:
3532:
3531:
3521:
3514:
3513:
3498:
3497:
3496:
3486:
3479:
3478:
3460:
3459:
3458:
3442:
3435:
3434:
3413:
3412:
3411:
3401:
3394:
3393:
3362:
3360:
3359:
3354:
3349:
3348:
3347:
3338:
3331:
3330:
3319:
3309:
3308:
3298:
3297:
3296:
3286:
3279:
3278:
3263:
3262:
3261:
3251:
3244:
3243:
3222:
3220:
3219:
3214:
3212:
3211:
3210:
3201:
3194:
3193:
3182:
3172:
3171:
3161:
3160:
3159:
3149:
3142:
3141:
3110:
3102:
3100:
3099:
3094:
3086:
3085:
3064:
3063:
3044:
3038:
3027:hexagonal number
3020:
3012:
3004:
2996:
2989:
2981:
2979:
2978:
2973:
2968:
2963:
2928:
2923:
2918:
2898:
2895:
2890:
2872:
2871:
2861:
2856:
2830:
2824:
2818:
2816:
2815:
2810:
2805:
2804:
2799:
2795:
2790:
2785:
2761:
2760:
2750:
2745:
2723:
2715:
2703:
2697:
2685:
2683:
2682:
2677:
2669:
2668:
2652:
2650:
2649:
2644:
2636:
2635:
2619:
2617:
2616:
2611:
2603:
2602:
2584:
2583:
2562:
2561:
2537:
2535:
2534:
2529:
2521:
2520:
2504:
2502:
2501:
2496:
2494:
2490:
2483:
2482:
2465:
2464:
2449:
2448:
2413:
2407:
2396:
2394:
2393:
2388:
2383:
2382:
2373:
2372:
2354:
2353:
2338:
2337:
2325:
2321:
2320:
2312:
2307:
2302:
2301:
2292:
2282:
2278:
2277:
2269:
2264:
2259:
2258:
2249:
2239:
2235:
2234:
2229:
2228:
2227:
2224:
2223:
2218:
2214:
2184:
2179:
2174:
2173:
2168:
2164:
2148:
2138:
2134:
2133:
2125:
2120:
2115:
2114:
2105:
2095:
2094:
2076:
2075:
2048:
2046:
2045:
2040:
2035:
2027:
2022:
2020:
2019:
2010:
2009:
2000:
1997:
1969:
1967:
1966:
1961:
1953:
1952:
1941:
1916:
1915:
1897:
1896:
1885:
1882:
1879:
1875:
1874:
1873:
1860:
1847:
1846:
1825:
1824:
1792:
1790:
1789:
1784:
1671:Termial function
1665:
1651:
1645:
1633:
1627:
1616:
1577:
1570:
1569:
1567:
1566:
1563:
1560:
1543:
1541:
1540:
1535:
1523:
1521:
1520:
1515:
1503:
1501:
1500:
1495:
1483:
1481:
1480:
1475:
1463:
1461:
1460:
1455:
1437:
1435:
1434:
1429:
1415:
1413:
1412:
1407:
1405:
1398:
1393:
1361:
1353:
1349:
1344:
1328:
1327:
1317:
1309:
1305:
1300:
1278:
1277:
1267:
1259:
1255:
1250:
1215:
1207:
1191:
1186:
1166:
1135:
1130:
1102:
1100:
1099:
1094:
1076:
1074:
1073:
1068:
1066:
1061:
1041:
1032:
1027:
1009:
1008:
992:
990:
989:
984:
970:
968:
967:
962:
954:
946:
941:
936:
916:
907:
902:
884:
883:
865:
863:
862:
857:
839:
833:
831:
830:
825:
817:
812:
792:
787:
786:
770:
768:
767:
762:
733:
732:
711:
709:
708:
703:
701:
700:
684:
682:
681:
676:
674:
669:
649:
644:
643:
627:
625:
624:
619:
589:
587:
586:
581:
579:
578:
558:
556:
555:
550:
545:
515:
513:
512:
507:
492:
486:
475:
473:
472:
467:
464:
463:
462:
453:
441:
424:
422:
421:
416:
413:
409:
408:
407:
398:
386:
373:
369:
364:
344:
339:
334:
333:
332:
304:
303:
293:
285:
253:
248:
226:
225:
193:
183:
173:
164:
155:
146:
137:
112:
92:
83:
77:
71:
9104:
9103:
9099:
9098:
9097:
9095:
9094:
9093:
9089:Simplex numbers
9049:
9048:
9047:
9042:
9020:
9016:Strobogrammatic
9007:
8989:
8971:
8953:
8935:
8917:
8899:
8881:
8858:
8837:
8821:
8780:Divisor-related
8775:
8735:
8686:
8656:
8593:
8577:
8556:
8523:
8496:
8484:
8466:
8378:
8377:related numbers
8351:
8328:
8295:
8286:Perfect totient
8252:
8229:
8160:Highly abundant
8102:
8081:
8013:
7996:
7968:
7951:
7937:Stirling second
7843:
7820:
7781:
7763:
7720:
7669:
7606:
7567:Centered square
7535:
7518:
7480:
7465:
7432:
7417:
7369:
7368:defined numbers
7351:
7318:
7303:
7274:Double Mersenne
7260:
7241:
7163:
7149:
7147:natural numbers
7143:
7113:
7108:
7090:
7047:
6996:Kinds of series
6987:
6926:
6893:Explicit series
6884:
6858:
6820:
6806:Cauchy sequence
6794:
6781:
6735:Figurate number
6712:
6706:
6697:Powers of three
6641:
6632:
6602:
6593:
6548:
6509:
6486:
6448:
6397:
6339:
6272:
6267:
6210:
6209:
6173:
6170:
6165:
6151:
6130:
6129:
6125:
6116:
6112:
6095:Euler, Leonhard
6093:
6092:
6085:
6070:
6047:
6046:
6042:
6035:
6012:
6011:
6007:
5997:
5995:
5989:
5988:
5984:
5974:
5972:
5962:10.2307/3621134
5943:
5942:
5938:
5922:
5921:
5917:
5879:
5878:
5874:
5867:
5863:
5856:
5852:
5812:
5805:
5801:
5800:
5796:
5787:Knuth, Donald.
5786:
5785:
5781:
5771:
5769:
5762:MathCircles.org
5756:
5755:
5751:
5742:
5738:
5733:
5729:
5719:
5717:
5711:
5710:
5706:
5697:
5695:
5681:
5680:
5676:
5669:
5653:Spivak, Michael
5651:
5650:
5646:
5633:
5632:
5628:
5624:
5600:
5589:
5583:
5572:
5563:
5559:
5555:
5551:
5543:
5536:
5490:
5477:
5476:
5469:
5461:
5455:
5451:
5447:
5440:
5364:
5210:
5185:
5182:
5179:
5178:
5176:
5169:
5166:
5163:
5162:
5160:
5153:
5150:
5147:
5146:
5144:
5143:in the second,
5137:
5134:
5131:
5130:
5128:
5121:
5118:
5115:
5114:
5112:
5106:
5100:
5094:
5088:
5082:
5075:
5067:
5064:
5055:
5054:
5052:
5042:
5036:
5033:
5025:
5011:
5002:
4996:
4986:
4977:
4971:
4944:
4926:
4925:
4920:
4915:
4910:
4905:
4898:
4897:
4892:
4887:
4882:
4877:
4872:
4866:
4864:
4859:
4854:
4849:
4836:
4835:
4808:
4795:
4782:
4777:
4776:
4773:Theon of Smyrna
4748:
4740:
4739:
4658:
4657:
4576:
4575:
4553:
4534:
4529:
4528:
4515:theta functions
4485:
4475:num = Δ + Δ + Δ
4474:
4468:
4462:
4429:
4413:
4400:
4390:
4374:
4369:
4368:
4338:
4325:
4306:
4301:
4300:
4228:
4227:
4185:
4128:
4094:
4093:
4088:The sum of the
4083:
4073:
4070:
4060:
4057:
4047:
4044:
4034:
4031:
4023:
4017:
4009:
4001:
3993:
3985:
3977:
3969:
3961:
3955:
3949:
3941:
3932:
3926:
3918:
3915:
3909:
3908:
3906:
3901:
3895:
3885:
3879:
3870:
3864:78 = 9 × 8 + 6
3861:66 = 9 × 7 + 3
3858:55 = 9 × 6 + 1
3849:28 = 9 × 3 + 1
3846:21 = 9 × 2 + 3
3843:15 = 9 × 1 + 6
3840:10 = 9 × 1 + 1
3802:
3797:
3774:
3769:
3741:
3728:
3727:
3706:
3696:
3691:
3690:
3680:
3679:
3678:
3673:
3668:
3667:
3660:
3655:
3638:
3557:
3556:
3549:
3535:
3523:
3505:
3488:
3470:
3444:
3420:
3403:
3379:
3369:
3368:
3322:
3321:
3314:
3300:
3288:
3270:
3253:
3235:
3225:
3224:
3185:
3184:
3177:
3163:
3151:
3133:
3123:
3122:
3106:
3071:
3055:
3047:
3046:
3040:
3034:
3014:
3006:
3000:
2992:
2985:
2929:
2899:
2863:
2837:
2836:
2826:
2822:
2770:
2766:
2765:
2752:
2726:
2725:
2719:
2711:
2699:
2693:
2660:
2655:
2654:
2627:
2622:
2621:
2588:
2569:
2553:
2548:
2547:
2512:
2507:
2506:
2474:
2470:
2466:
2456:
2434:
2429:
2428:
2414:
2374:
2358:
2345:
2329:
2293:
2290:
2286:
2250:
2247:
2243:
2204:
2200:
2199:
2185:
2154:
2150:
2149:
2146:
2142:
2106:
2103:
2099:
2080:
2067:
2062:
2061:
2054:
2011:
2001:
1978:
1977:
1944:
1901:
1888:
1855:
1832:
1816:
1811:
1810:
1804:
1706:
1705:
1661:
1647:
1646:. The function
1644:
1635:
1629:
1622:
1615:
1607:
1572:
1564:
1561:
1556:
1555:
1553:
1552:
1526:
1525:
1506:
1505:
1486:
1485:
1466:
1465:
1440:
1439:
1420:
1419:
1403:
1402:
1362:
1351:
1350:
1319:
1318:
1307:
1306:
1269:
1268:
1257:
1256:
1216:
1205:
1204:
1167:
1158:
1107:
1106:
1103:to this yields
1079:
1078:
1042:
1000:
995:
994:
975:
974:
917:
875:
870:
869:
848:
847:
840:
793:
778:
773:
772:
724:
716:
715:
692:
687:
686:
685:. The example
650:
635:
630:
629:
592:
591:
570:
565:
564:
518:
517:
498:
497:
488:
481:
443:
436:
429:
428:
425:
411:
410:
388:
381:
371:
370:
345:
295:
294:
283:
282:
227:
217:
207:
206:
199:
191:
189:
181:
179:
171:
169:
162:
160:
153:
151:
144:
142:
140:Natural numbers
135:
123:
108:
105:
88:
85:natural numbers
79:
73:
67:
56:figurate number
48:triangle number
29:
17:
16:Figurate number
12:
11:
5:
9102:
9100:
9092:
9091:
9086:
9081:
9076:
9071:
9066:
9061:
9051:
9050:
9044:
9043:
9041:
9040:
9029:
9026:
9025:
9022:
9021:
9019:
9018:
9012:
9009:
9008:
9002:
8995:
8994:
8991:
8990:
8988:
8987:
8982:
8976:
8973:
8972:
8966:
8959:
8958:
8955:
8954:
8952:
8951:
8949:Sorting number
8946:
8944:Pancake number
8940:
8937:
8936:
8930:
8923:
8922:
8919:
8918:
8916:
8915:
8910:
8904:
8901:
8900:
8894:
8887:
8886:
8883:
8882:
8880:
8879:
8874:
8869:
8863:
8860:
8859:
8856:Binary numbers
8854:
8847:
8846:
8843:
8842:
8839:
8838:
8836:
8835:
8829:
8827:
8823:
8822:
8820:
8819:
8814:
8809:
8804:
8799:
8794:
8789:
8783:
8781:
8777:
8776:
8774:
8773:
8768:
8763:
8758:
8753:
8747:
8745:
8737:
8736:
8734:
8733:
8728:
8723:
8718:
8713:
8708:
8703:
8697:
8695:
8688:
8687:
8685:
8684:
8683:
8682:
8671:
8669:
8666:P-adic numbers
8662:
8661:
8658:
8657:
8655:
8654:
8653:
8652:
8642:
8637:
8632:
8627:
8622:
8617:
8612:
8607:
8601:
8599:
8595:
8594:
8592:
8591:
8585:
8583:
8582:Coding-related
8579:
8578:
8576:
8575:
8570:
8564:
8562:
8558:
8557:
8555:
8554:
8549:
8544:
8539:
8533:
8531:
8522:
8521:
8520:
8519:
8517:Multiplicative
8514:
8503:
8501:
8486:
8485:
8481:Numeral system
8479:
8472:
8471:
8468:
8467:
8465:
8464:
8459:
8454:
8449:
8444:
8439:
8434:
8429:
8424:
8419:
8414:
8409:
8404:
8399:
8394:
8389:
8383:
8380:
8379:
8368:
8361:
8360:
8357:
8356:
8353:
8352:
8350:
8349:
8344:
8338:
8336:
8330:
8329:
8327:
8326:
8321:
8316:
8311:
8305:
8303:
8297:
8296:
8294:
8293:
8288:
8283:
8278:
8273:
8271:Highly totient
8268:
8262:
8260:
8254:
8253:
8251:
8250:
8245:
8239:
8237:
8231:
8230:
8228:
8227:
8222:
8217:
8212:
8207:
8202:
8197:
8192:
8187:
8182:
8177:
8172:
8167:
8162:
8157:
8152:
8147:
8142:
8137:
8132:
8127:
8125:Almost perfect
8122:
8116:
8114:
8104:
8103:
8094:
8087:
8086:
8083:
8082:
8080:
8079:
8074:
8069:
8064:
8059:
8054:
8049:
8044:
8039:
8034:
8029:
8024:
8018:
8015:
8014:
8009:
8002:
8001:
7998:
7997:
7995:
7994:
7989:
7984:
7979:
7973:
7970:
7969:
7964:
7957:
7956:
7953:
7952:
7950:
7949:
7944:
7939:
7934:
7932:Stirling first
7929:
7924:
7919:
7914:
7909:
7904:
7899:
7894:
7889:
7884:
7879:
7874:
7869:
7864:
7859:
7854:
7848:
7845:
7844:
7841:
7834:
7833:
7830:
7829:
7826:
7825:
7822:
7821:
7819:
7818:
7813:
7808:
7802:
7800:
7793:
7787:
7786:
7783:
7782:
7780:
7779:
7773:
7771:
7765:
7764:
7762:
7761:
7756:
7751:
7746:
7741:
7736:
7730:
7728:
7722:
7721:
7719:
7718:
7713:
7708:
7703:
7698:
7692:
7690:
7681:
7675:
7674:
7671:
7670:
7668:
7667:
7662:
7657:
7652:
7647:
7642:
7637:
7632:
7627:
7622:
7616:
7614:
7608:
7607:
7605:
7604:
7599:
7594:
7589:
7584:
7579:
7574:
7569:
7564:
7558:
7556:
7547:
7537:
7536:
7531:
7524:
7523:
7520:
7519:
7517:
7516:
7511:
7506:
7501:
7496:
7491:
7485:
7482:
7481:
7478:
7471:
7470:
7467:
7466:
7464:
7463:
7458:
7453:
7448:
7443:
7437:
7434:
7433:
7430:
7423:
7422:
7419:
7418:
7416:
7415:
7410:
7405:
7400:
7395:
7390:
7385:
7380:
7374:
7371:
7370:
7364:
7357:
7356:
7353:
7352:
7350:
7349:
7344:
7339:
7334:
7329:
7323:
7320:
7319:
7316:
7309:
7308:
7305:
7304:
7302:
7301:
7296:
7291:
7286:
7281:
7276:
7271:
7265:
7262:
7261:
7254:
7247:
7246:
7243:
7242:
7240:
7239:
7234:
7229:
7224:
7219:
7214:
7209:
7204:
7199:
7194:
7189:
7184:
7179:
7174:
7168:
7165:
7164:
7158:
7151:
7150:
7144:
7142:
7141:
7134:
7127:
7119:
7110:
7109:
7107:
7106:
7095:
7092:
7091:
7089:
7088:
7083:
7078:
7073:
7068:
7063:
7057:
7055:
7049:
7048:
7046:
7045:
7040:
7038:Fourier series
7035:
7030:
7025:
7023:Puiseux series
7020:
7018:Laurent series
7015:
7010:
7005:
6999:
6997:
6993:
6992:
6989:
6988:
6986:
6985:
6980:
6975:
6970:
6965:
6960:
6955:
6950:
6945:
6940:
6934:
6932:
6928:
6927:
6925:
6924:
6919:
6914:
6909:
6903:
6901:
6894:
6890:
6889:
6886:
6885:
6883:
6882:
6877:
6872:
6866:
6864:
6860:
6859:
6857:
6856:
6851:
6846:
6841:
6835:
6833:
6826:
6822:
6821:
6819:
6818:
6813:
6808:
6802:
6800:
6796:
6795:
6788:
6786:
6783:
6782:
6780:
6779:
6778:
6777:
6767:
6762:
6757:
6752:
6747:
6742:
6737:
6732:
6727:
6721:
6719:
6708:
6707:
6705:
6704:
6699:
6694:
6689:
6684:
6679:
6674:
6669:
6664:
6658:
6656:
6649:
6643:
6642:
6633:
6631:
6630:
6623:
6616:
6608:
6599:
6598:
6595:
6594:
6592:
6591:
6586:
6581:
6576:
6570:
6568:
6561:
6554:
6553:
6550:
6549:
6547:
6546:
6541:
6536:
6530:
6528:
6521:
6515:
6514:
6511:
6510:
6508:
6507:
6502:
6496:
6494:
6488:
6487:
6485:
6484:
6479:
6474:
6469:
6464:
6458:
6456:
6450:
6449:
6447:
6446:
6441:
6436:
6431:
6426:
6420:
6418:
6409:
6403:
6402:
6399:
6398:
6396:
6395:
6390:
6385:
6380:
6375:
6370:
6365:
6360:
6358:Square numbers
6355:
6349:
6347:
6341:
6340:
6338:
6337:
6332:
6327:
6322:
6317:
6312:
6307:
6302:
6297:
6291:
6289:
6280:
6274:
6273:
6268:
6266:
6265:
6258:
6251:
6243:
6237:
6236:
6226:
6207:
6198:
6189:
6169:
6168:External links
6166:
6164:
6163:
6149:
6123:
6110:
6083:
6068:
6040:
6033:
6005:
5982:
5936:
5915:
5888:(4): 407–434.
5872:
5861:
5850:
5794:
5779:
5749:
5736:
5727:
5712:Eves, Howard.
5704:
5682:Hayes, Brian.
5674:
5667:
5644:
5625:
5623:
5620:
5619:
5618:
5612:
5606:
5599:
5596:
5571:
5568:
5541:if and only if
5539:is triangular
5516:
5512:
5509:
5504:
5501:
5498:
5495:
5487:
5484:
5465:
5454:as the number
5439:
5436:
5433:
5432:
5429:
5426:
5423:
5420:
5417:
5414:
5411:
5408:
5405:
5402:
5399:
5396:
5393:
5390:
5387:
5384:
5381:
5378:
5375:
5372:
5369:
5366:
5362:
5357:
5356:
5353:
5350:
5347:
5344:
5341:
5338:
5335:
5332:
5329:
5326:
5323:
5320:
5317:
5314:
5311:
5308:
5305:
5302:
5299:
5296:
5293:
5290:
5284:
5283:
5280:
5277:
5274:
5271:
5268:
5265:
5262:
5259:
5256:
5253:
5250:
5247:
5244:
5241:
5238:
5235:
5232:
5229:
5226:
5223:
5220:
5217:
5199:game designers
5071:
5029:
5006:
4981:
4943:
4940:
4924:
4921:
4919:
4916:
4914:
4911:
4909:
4906:
4904:
4901:
4899:
4896:
4893:
4891:
4888:
4886:
4883:
4881:
4878:
4876:
4873:
4871:
4868:
4867:
4863:
4860:
4858:
4855:
4853:
4850:
4848:
4845:
4843:
4815:
4811:
4807:
4802:
4798:
4794:
4789:
4785:
4769:
4768:
4755:
4751:
4747:
4737:
4724:
4719:
4716:
4713:
4710:
4707:
4704:
4701:
4698:
4695:
4692:
4689:
4684:
4679:
4673:
4670:
4665:
4655:
4644:
4641:
4638:
4635:
4632:
4629:
4623:
4620:
4615:
4612:
4609:
4606:
4603:
4600:
4597:
4591:
4588:
4583:
4573:
4560:
4556:
4552:
4547:
4544:
4541:
4537:
4466:
4447:
4442:
4439:
4436:
4432:
4426:
4423:
4420:
4416:
4412:
4407:
4403:
4397:
4393:
4389:
4384:
4381:
4377:
4356:
4353:
4350:
4345:
4341:
4337:
4332:
4328:
4324:
4319:
4316:
4313:
4309:
4286:
4283:
4277:
4274:
4271:
4268:
4265:
4262:
4258:
4251:
4246:
4243:
4240:
4236:
4209:
4206:
4200:
4197:
4192:
4188:
4183:
4176:
4171:
4168:
4165:
4161:
4157:
4154:
4147:
4143:
4140:
4135:
4131:
4124:
4117:
4112:
4109:
4106:
4102:
4077:
4064:
4051:
4038:
4027:
3937:
3837:6 = 9 × 0 + 6
3834:3 = 9 × 0 + 3
3831:1 = 9 × 0 + 1
3828:
3807:Mersenne prime
3800:
3781:
3777:
3772:
3768:
3763:
3759:
3756:
3753:
3748:
3744:
3740:
3735:
3731:
3724:
3719:
3716:
3713:
3709:
3703:
3699:
3687:perfect number
3670:
3669:
3661:
3656:
3649:
3648:
3647:
3646:
3637:
3634:
3631:, and so forth
3587:
3582:
3578:
3575:
3572:
3569:
3564:
3560:
3553:
3547:
3542:
3538:
3530:
3526:
3520:
3517:
3512:
3508:
3503:
3495:
3491:
3485:
3482:
3477:
3473:
3468:
3464:
3457:
3454:
3451:
3447:
3441:
3438:
3433:
3430:
3427:
3423:
3418:
3410:
3406:
3400:
3397:
3392:
3389:
3386:
3382:
3377:
3352:
3346:
3341:
3337:
3334:
3329:
3325:
3318:
3312:
3307:
3303:
3295:
3291:
3285:
3282:
3277:
3273:
3268:
3260:
3256:
3250:
3247:
3242:
3238:
3233:
3209:
3204:
3200:
3197:
3192:
3188:
3181:
3175:
3170:
3166:
3158:
3154:
3148:
3145:
3140:
3136:
3131:
3092:
3089:
3084:
3081:
3078:
3074:
3070:
3067:
3062:
3058:
3054:
2971:
2966:
2962:
2959:
2956:
2953:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2926:
2921:
2917:
2914:
2911:
2908:
2905:
2902:
2894:
2889:
2886:
2883:
2879:
2875:
2870:
2866:
2860:
2855:
2852:
2849:
2845:
2808:
2803:
2798:
2794:
2789:
2784:
2781:
2778:
2774:
2769:
2764:
2759:
2755:
2749:
2744:
2741:
2738:
2734:
2710:square of the
2675:
2672:
2667:
2663:
2642:
2639:
2634:
2630:
2609:
2606:
2601:
2598:
2595:
2591:
2587:
2582:
2579:
2576:
2572:
2568:
2565:
2560:
2556:
2527:
2524:
2519:
2515:
2493:
2489:
2486:
2481:
2477:
2473:
2469:
2463:
2459:
2455:
2452:
2447:
2444:
2441:
2437:
2420:, is called a
2418:§ Formula
2401:
2386:
2381:
2377:
2371:
2368:
2365:
2361:
2357:
2352:
2348:
2344:
2341:
2336:
2332:
2328:
2324:
2318:
2315:
2310:
2305:
2300:
2296:
2289:
2285:
2281:
2275:
2272:
2267:
2262:
2257:
2253:
2246:
2242:
2238:
2232:
2222:
2217:
2213:
2210:
2207:
2203:
2194:
2191:
2188:
2182:
2177:
2172:
2167:
2163:
2160:
2157:
2153:
2145:
2141:
2137:
2131:
2128:
2123:
2118:
2113:
2109:
2102:
2098:
2093:
2090:
2087:
2083:
2079:
2074:
2070:
2053:
2050:
2038:
2033:
2030:
2025:
2018:
2014:
2008:
2004:
1996:
1993:
1990:
1986:
1959:
1956:
1951:
1947:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1914:
1911:
1908:
1904:
1900:
1895:
1891:
1878:
1872:
1867:
1864:
1859:
1853:
1850:
1845:
1842:
1839:
1835:
1831:
1828:
1823:
1819:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1734:
1731:
1728:
1725:
1722:
1719:
1716:
1713:
1639:
1611:
1544:by induction.
1533:
1513:
1493:
1473:
1453:
1450:
1447:
1427:
1401:
1396:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1359:
1356:
1354:
1352:
1347:
1343:
1340:
1337:
1334:
1331:
1326:
1322:
1315:
1312:
1310:
1308:
1303:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1276:
1272:
1265:
1262:
1260:
1258:
1253:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1213:
1210:
1208:
1206:
1203:
1200:
1197:
1194:
1189:
1185:
1182:
1179:
1176:
1173:
1170:
1164:
1161:
1159:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1134:
1129:
1126:
1123:
1119:
1115:
1114:
1092:
1089:
1086:
1064:
1060:
1057:
1054:
1051:
1048:
1045:
1039:
1036:
1031:
1026:
1023:
1020:
1016:
1012:
1007:
1003:
982:
960:
957:
952:
949:
944:
939:
935:
932:
929:
926:
923:
920:
914:
911:
906:
901:
898:
895:
891:
887:
882:
878:
855:
823:
820:
815:
811:
808:
805:
802:
799:
796:
790:
785:
781:
760:
757:
754:
751:
748:
745:
742:
739:
736:
731:
727:
723:
714:
699:
695:
672:
668:
665:
662:
659:
656:
653:
647:
642:
638:
617:
614:
611:
608:
605:
602:
599:
577:
573:
548:
544:
540:
537:
534:
531:
528:
525:
505:
461:
456:
452:
449:
446:
440:
406:
401:
397:
394:
391:
385:
379:
376:
374:
372:
367:
363:
360:
357:
354:
351:
348:
342:
337:
329:
326:
323:
320:
317:
310:
307:
302:
298:
291:
288:
286:
284:
281:
278:
275:
272:
269:
266:
263:
260:
257:
252:
247:
244:
241:
237:
233:
230:
228:
224:
220:
216:
215:
205:
190:
180:
170:
161:
152:
143:
134:
122:
119:
103:
60:square numbers
27:
15:
13:
10:
9:
6:
4:
3:
2:
9101:
9090:
9087:
9085:
9082:
9080:
9077:
9075:
9072:
9070:
9067:
9065:
9062:
9060:
9057:
9056:
9054:
9039:
9035:
9031:
9030:
9027:
9017:
9014:
9013:
9010:
9005:
9000:
8996:
8986:
8983:
8981:
8978:
8977:
8974:
8969:
8964:
8960:
8950:
8947:
8945:
8942:
8941:
8938:
8933:
8928:
8924:
8914:
8911:
8909:
8906:
8905:
8902:
8898:
8892:
8888:
8878:
8875:
8873:
8870:
8868:
8865:
8864:
8861:
8857:
8852:
8848:
8834:
8831:
8830:
8828:
8824:
8818:
8815:
8813:
8810:
8808:
8807:Polydivisible
8805:
8803:
8800:
8798:
8795:
8793:
8790:
8788:
8785:
8784:
8782:
8778:
8772:
8769:
8767:
8764:
8762:
8759:
8757:
8754:
8752:
8749:
8748:
8746:
8743:
8738:
8732:
8729:
8727:
8724:
8722:
8719:
8717:
8714:
8712:
8709:
8707:
8704:
8702:
8699:
8698:
8696:
8693:
8689:
8681:
8678:
8677:
8676:
8673:
8672:
8670:
8667:
8663:
8651:
8648:
8647:
8646:
8643:
8641:
8638:
8636:
8633:
8631:
8628:
8626:
8623:
8621:
8618:
8616:
8613:
8611:
8608:
8606:
8603:
8602:
8600:
8596:
8590:
8587:
8586:
8584:
8580:
8574:
8571:
8569:
8566:
8565:
8563:
8561:Digit product
8559:
8553:
8550:
8548:
8545:
8543:
8540:
8538:
8535:
8534:
8532:
8530:
8526:
8518:
8515:
8513:
8510:
8509:
8508:
8505:
8504:
8502:
8500:
8495:
8491:
8487:
8482:
8477:
8473:
8463:
8460:
8458:
8455:
8453:
8450:
8448:
8445:
8443:
8440:
8438:
8435:
8433:
8430:
8428:
8425:
8423:
8420:
8418:
8415:
8413:
8410:
8408:
8405:
8403:
8400:
8398:
8397:Erdős–Nicolas
8395:
8393:
8390:
8388:
8385:
8384:
8381:
8376:
8372:
8366:
8362:
8348:
8345:
8343:
8340:
8339:
8337:
8335:
8331:
8325:
8322:
8320:
8317:
8315:
8312:
8310:
8307:
8306:
8304:
8302:
8298:
8292:
8289:
8287:
8284:
8282:
8279:
8277:
8274:
8272:
8269:
8267:
8264:
8263:
8261:
8259:
8255:
8249:
8246:
8244:
8241:
8240:
8238:
8236:
8232:
8226:
8223:
8221:
8218:
8216:
8215:Superabundant
8213:
8211:
8208:
8206:
8203:
8201:
8198:
8196:
8193:
8191:
8188:
8186:
8183:
8181:
8178:
8176:
8173:
8171:
8168:
8166:
8163:
8161:
8158:
8156:
8153:
8151:
8148:
8146:
8143:
8141:
8138:
8136:
8133:
8131:
8128:
8126:
8123:
8121:
8118:
8117:
8115:
8113:
8109:
8105:
8101:
8097:
8092:
8088:
8078:
8075:
8073:
8070:
8068:
8065:
8063:
8060:
8058:
8055:
8053:
8050:
8048:
8045:
8043:
8040:
8038:
8035:
8033:
8030:
8028:
8025:
8023:
8020:
8019:
8016:
8012:
8007:
8003:
7993:
7990:
7988:
7985:
7983:
7980:
7978:
7975:
7974:
7971:
7967:
7962:
7958:
7948:
7945:
7943:
7940:
7938:
7935:
7933:
7930:
7928:
7925:
7923:
7920:
7918:
7915:
7913:
7910:
7908:
7905:
7903:
7900:
7898:
7895:
7893:
7890:
7888:
7885:
7883:
7880:
7878:
7875:
7873:
7870:
7868:
7865:
7863:
7860:
7858:
7855:
7853:
7850:
7849:
7846:
7839:
7835:
7817:
7814:
7812:
7809:
7807:
7804:
7803:
7801:
7797:
7794:
7792:
7791:4-dimensional
7788:
7778:
7775:
7774:
7772:
7770:
7766:
7760:
7757:
7755:
7752:
7750:
7747:
7745:
7742:
7740:
7737:
7735:
7732:
7731:
7729:
7727:
7723:
7717:
7714:
7712:
7709:
7707:
7704:
7702:
7701:Centered cube
7699:
7697:
7694:
7693:
7691:
7689:
7685:
7682:
7680:
7679:3-dimensional
7676:
7666:
7663:
7661:
7658:
7656:
7653:
7651:
7648:
7646:
7643:
7641:
7638:
7636:
7633:
7631:
7628:
7626:
7623:
7621:
7618:
7617:
7615:
7613:
7609:
7603:
7600:
7598:
7595:
7593:
7590:
7588:
7585:
7583:
7580:
7578:
7575:
7573:
7570:
7568:
7565:
7563:
7560:
7559:
7557:
7555:
7551:
7548:
7546:
7545:2-dimensional
7542:
7538:
7534:
7529:
7525:
7515:
7512:
7510:
7507:
7505:
7502:
7500:
7497:
7495:
7492:
7490:
7489:Nonhypotenuse
7487:
7486:
7483:
7476:
7472:
7462:
7459:
7457:
7454:
7452:
7449:
7447:
7444:
7442:
7439:
7438:
7435:
7428:
7424:
7414:
7411:
7409:
7406:
7404:
7401:
7399:
7396:
7394:
7391:
7389:
7386:
7384:
7381:
7379:
7376:
7375:
7372:
7367:
7362:
7358:
7348:
7345:
7343:
7340:
7338:
7335:
7333:
7330:
7328:
7325:
7324:
7321:
7314:
7310:
7300:
7297:
7295:
7292:
7290:
7287:
7285:
7282:
7280:
7277:
7275:
7272:
7270:
7267:
7266:
7263:
7258:
7252:
7248:
7238:
7235:
7233:
7230:
7228:
7227:Perfect power
7225:
7223:
7220:
7218:
7217:Seventh power
7215:
7213:
7210:
7208:
7205:
7203:
7200:
7198:
7195:
7193:
7190:
7188:
7185:
7183:
7180:
7178:
7175:
7173:
7170:
7169:
7166:
7161:
7156:
7152:
7148:
7140:
7135:
7133:
7128:
7126:
7121:
7120:
7117:
7105:
7097:
7096:
7093:
7087:
7084:
7082:
7079:
7077:
7074:
7072:
7069:
7067:
7064:
7062:
7059:
7058:
7056:
7054:
7050:
7044:
7041:
7039:
7036:
7034:
7031:
7029:
7026:
7024:
7021:
7019:
7016:
7014:
7011:
7009:
7006:
7004:
7003:Taylor series
7001:
7000:
6998:
6994:
6984:
6981:
6979:
6976:
6974:
6971:
6969:
6966:
6964:
6961:
6959:
6956:
6954:
6951:
6949:
6946:
6944:
6941:
6939:
6936:
6935:
6933:
6929:
6923:
6920:
6918:
6915:
6913:
6910:
6908:
6905:
6904:
6902:
6898:
6895:
6891:
6881:
6878:
6876:
6873:
6871:
6868:
6867:
6865:
6861:
6855:
6852:
6850:
6847:
6845:
6842:
6840:
6837:
6836:
6834:
6830:
6827:
6823:
6817:
6814:
6812:
6809:
6807:
6804:
6803:
6801:
6797:
6792:
6776:
6773:
6772:
6771:
6768:
6766:
6763:
6761:
6758:
6756:
6753:
6751:
6748:
6746:
6743:
6741:
6738:
6736:
6733:
6731:
6728:
6726:
6723:
6722:
6720:
6716:
6709:
6703:
6700:
6698:
6695:
6693:
6692:Powers of two
6690:
6688:
6685:
6683:
6680:
6678:
6677:Square number
6675:
6673:
6670:
6668:
6665:
6663:
6660:
6659:
6657:
6653:
6650:
6648:
6644:
6640:
6636:
6629:
6624:
6622:
6617:
6615:
6610:
6609:
6606:
6590:
6587:
6585:
6582:
6580:
6577:
6575:
6572:
6571:
6569:
6565:
6562:
6560:
6555:
6545:
6542:
6540:
6537:
6535:
6532:
6531:
6529:
6525:
6522:
6520:
6519:4-dimensional
6516:
6506:
6503:
6501:
6498:
6497:
6495:
6493:
6489:
6483:
6480:
6478:
6475:
6473:
6470:
6468:
6465:
6463:
6460:
6459:
6457:
6455:
6451:
6445:
6442:
6440:
6437:
6435:
6432:
6430:
6427:
6425:
6422:
6421:
6419:
6417:
6413:
6410:
6408:
6407:3-dimensional
6404:
6394:
6391:
6389:
6386:
6384:
6381:
6379:
6376:
6374:
6371:
6369:
6366:
6364:
6361:
6359:
6356:
6354:
6351:
6350:
6348:
6346:
6342:
6336:
6333:
6331:
6328:
6326:
6323:
6321:
6318:
6316:
6313:
6311:
6308:
6306:
6303:
6301:
6298:
6296:
6293:
6292:
6290:
6288:
6284:
6281:
6279:
6278:2-dimensional
6275:
6271:
6264:
6259:
6257:
6252:
6250:
6245:
6244:
6241:
6234:
6230:
6227:
6222:
6221:
6216:
6213:
6208:
6206:
6202:
6199:
6197:
6193:
6190:
6186:
6182:
6181:
6176:
6172:
6171:
6167:
6160:
6156:
6152:
6146:
6142:
6138:
6134:
6127:
6124:
6120:
6114:
6111:
6106:
6105:
6100:
6096:
6090:
6088:
6084:
6079:
6075:
6071:
6065:
6061:
6057:
6053:
6052:
6044:
6041:
6036:
6034:9781421417288
6030:
6026:
6022:
6018:
6017:
6009:
6006:
5993:
5986:
5983:
5971:
5967:
5963:
5959:
5955:
5951:
5947:
5940:
5937:
5931:
5926:
5919:
5916:
5911:
5907:
5903:
5899:
5895:
5891:
5887:
5883:
5876:
5873:
5870:
5865:
5862:
5859:
5854:
5851:
5846:
5842:
5838:
5834:
5830:
5826:
5822:
5819:(in German).
5818:
5811:
5808:
5798:
5795:
5790:
5783:
5780:
5767:
5763:
5759:
5753:
5750:
5746:
5740:
5737:
5731:
5728:
5716:. Mathcentral
5715:
5708:
5705:
5694:on 2015-04-02
5693:
5689:
5685:
5678:
5675:
5670:
5664:
5660:
5659:
5654:
5648:
5645:
5640:
5636:
5630:
5627:
5621:
5616:
5613:
5610:
5607:
5605:
5602:
5601:
5597:
5595:
5592:
5586:
5581:
5577:
5569:
5567:
5547:
5542:
5534:
5529:
5514:
5510:
5507:
5502:
5499:
5496:
5493:
5485:
5482:
5473:
5468:
5464:
5458:
5445:
5437:
5430:
5427:
5424:
5421:
5418:
5415:
5412:
5409:
5406:
5403:
5400:
5397:
5394:
5391:
5388:
5385:
5382:
5379:
5376:
5373:
5370:
5367:
5365:
5359:
5358:
5354:
5351:
5348:
5345:
5342:
5339:
5336:
5333:
5330:
5327:
5324:
5321:
5318:
5315:
5312:
5309:
5306:
5303:
5300:
5297:
5294:
5291:
5289:
5286:
5285:
5281:
5278:
5275:
5272:
5269:
5266:
5263:
5260:
5257:
5254:
5251:
5248:
5245:
5242:
5239:
5236:
5233:
5230:
5227:
5224:
5221:
5218:
5215:
5214:
5208:
5202:
5200:
5196:
5192:
5109:
5103:
5097:
5091:
5085:
5074:
5070:
5062:
5058:
5050:
5046:
5039:
5032:
5028:
5023:
5019:
5014:
5009:
5005:
4999:
4994:
4989:
4984:
4980:
4974:
4969:
4961:
4957:
4953:
4948:
4941:
4939:
4922:
4917:
4912:
4907:
4902:
4894:
4889:
4884:
4879:
4874:
4869:
4861:
4856:
4851:
4846:
4833:
4831:
4813:
4809:
4805:
4800:
4796:
4792:
4787:
4783:
4774:
4753:
4749:
4745:
4738:
4714:
4711:
4708:
4702:
4696:
4693:
4690:
4677:
4671:
4668:
4663:
4656:
4639:
4636:
4633:
4627:
4621:
4618:
4613:
4607:
4604:
4601:
4595:
4589:
4586:
4581:
4574:
4558:
4554:
4550:
4545:
4542:
4539:
4535:
4527:
4526:
4525:
4522:
4520:
4516:
4511:
4509:
4505:
4501:
4497:
4495:
4491:
4482:
4480:
4473:
4465:
4459:
4445:
4440:
4437:
4434:
4430:
4424:
4421:
4418:
4414:
4410:
4405:
4401:
4395:
4391:
4387:
4382:
4379:
4375:
4354:
4351:
4348:
4343:
4339:
4335:
4330:
4326:
4322:
4317:
4314:
4311:
4307:
4297:
4284:
4281:
4272:
4269:
4266:
4260:
4256:
4244:
4241:
4238:
4234:
4225:
4220:
4207:
4204:
4198:
4195:
4190:
4186:
4181:
4169:
4166:
4163:
4159:
4155:
4152:
4145:
4141:
4138:
4133:
4129:
4122:
4110:
4107:
4104:
4100:
4091:
4086:
4085:, and so on.
4081:
4076:
4068:
4063:
4055:
4050:
4042:
4037:
4030:
4026:
4020:
4013:
4005:
3997:
3989:
3981:
3973:
3965:
3958:
3952:
3945:
3940:
3936:
3929:
3912:
3904:
3898:
3892:
3888:
3882:
3876:
3873:
3868:
3865:
3862:
3859:
3856:
3853:
3850:
3847:
3844:
3841:
3838:
3835:
3832:
3827:
3825:
3821:
3816:
3813:
3810:
3808:
3803:
3779:
3775:
3770:
3766:
3761:
3754:
3751:
3746:
3742:
3733:
3729:
3722:
3717:
3714:
3711:
3707:
3701:
3697:
3688:
3683:
3676:
3672:
3665:
3659:
3653:
3645:
3643:
3635:
3630:
3626:
3622:
3618:
3614:
3610:
3605:
3601:
3580:
3576:
3573:
3570:
3567:
3562:
3558:
3545:
3540:
3536:
3528:
3524:
3518:
3515:
3510:
3506:
3501:
3493:
3489:
3483:
3480:
3475:
3471:
3466:
3462:
3455:
3452:
3449:
3445:
3439:
3436:
3431:
3428:
3425:
3421:
3416:
3408:
3404:
3398:
3395:
3390:
3387:
3384:
3380:
3375:
3366:
3350:
3339:
3335:
3332:
3327:
3323:
3310:
3305:
3301:
3293:
3289:
3283:
3280:
3275:
3271:
3266:
3258:
3254:
3248:
3245:
3240:
3236:
3231:
3202:
3198:
3195:
3190:
3186:
3173:
3168:
3164:
3156:
3152:
3146:
3143:
3138:
3134:
3129:
3119:
3117:
3112:
3109:
3103:
3090:
3087:
3082:
3079:
3076:
3072:
3068:
3065:
3060:
3056:
3052:
3043:
3037:
3032:
3028:
3024:
3018:
3010:
3003:
2998:
2997:-gonal number
2995:
2988:
2982:
2969:
2964:
2957:
2954:
2951:
2942:
2939:
2936:
2930:
2924:
2919:
2912:
2909:
2906:
2900:
2892:
2887:
2884:
2881:
2877:
2873:
2868:
2864:
2858:
2853:
2850:
2847:
2843:
2834:
2829:
2819:
2806:
2801:
2796:
2792:
2787:
2782:
2779:
2776:
2772:
2767:
2762:
2757:
2753:
2747:
2742:
2739:
2736:
2732:
2722:
2717:
2714:
2704:cube numbers.
2702:
2696:
2690:
2686:
2673:
2670:
2665:
2661:
2640:
2637:
2632:
2628:
2607:
2604:
2599:
2596:
2593:
2589:
2585:
2580:
2577:
2574:
2570:
2566:
2563:
2558:
2554:
2545:
2542:
2538:
2525:
2522:
2517:
2513:
2491:
2487:
2484:
2479:
2475:
2471:
2467:
2461:
2457:
2453:
2450:
2445:
2442:
2439:
2435:
2425:
2423:
2422:pronic number
2419:
2412:
2406:
2400:
2397:
2384:
2379:
2369:
2366:
2363:
2359:
2355:
2350:
2346:
2339:
2334:
2330:
2326:
2322:
2316:
2313:
2308:
2303:
2298:
2294:
2287:
2283:
2279:
2273:
2270:
2265:
2260:
2255:
2251:
2244:
2240:
2236:
2230:
2220:
2215:
2211:
2208:
2205:
2201:
2192:
2189:
2186:
2180:
2175:
2170:
2165:
2161:
2158:
2155:
2151:
2143:
2139:
2135:
2129:
2126:
2121:
2116:
2111:
2107:
2100:
2096:
2091:
2088:
2085:
2081:
2077:
2072:
2068:
2057:
2051:
2049:
2036:
2031:
2028:
2023:
2016:
2012:
2006:
2002:
1988:
1975:
1970:
1957:
1954:
1949:
1945:
1938:
1932:
1929:
1926:
1920:
1917:
1912:
1909:
1906:
1902:
1898:
1893:
1889:
1876:
1865:
1862:
1851:
1848:
1843:
1840:
1837:
1833:
1829:
1826:
1821:
1817:
1808:
1802:
1800:
1799:
1793:
1780:
1777:
1774:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1703:
1701:
1700:
1696:For example,
1694:
1692:
1691:
1686:
1685:
1680:
1676:
1672:
1667:
1664:
1659:
1655:
1650:
1642:
1638:
1632:
1625:
1620:
1614:
1610:
1601:
1597:
1593:
1591:
1589:
1585:
1581:
1575:
1559:
1550:
1545:
1531:
1511:
1491:
1471:
1451:
1448:
1445:
1425:
1416:
1399:
1394:
1387:
1384:
1381:
1372:
1369:
1366:
1357:
1355:
1345:
1341:
1338:
1335:
1332:
1329:
1324:
1320:
1313:
1311:
1301:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1274:
1270:
1263:
1261:
1251:
1247:
1244:
1241:
1238:
1235:
1229:
1226:
1223:
1217:
1211:
1209:
1201:
1198:
1195:
1192:
1187:
1180:
1177:
1174:
1168:
1162:
1160:
1152:
1149:
1146:
1140:
1137:
1132:
1127:
1124:
1121:
1117:
1104:
1090:
1087:
1084:
1062:
1055:
1052:
1049:
1043:
1037:
1034:
1029:
1024:
1021:
1018:
1014:
1010:
1005:
1001:
980:
971:
958:
955:
950:
947:
942:
937:
930:
927:
924:
918:
912:
909:
904:
899:
896:
893:
889:
885:
880:
876:
867:
853:
845:
838:
821:
818:
813:
806:
803:
800:
794:
788:
783:
779:
758:
755:
749:
746:
743:
737:
734:
729:
725:
721:
713:
697:
693:
670:
663:
660:
657:
651:
645:
640:
636:
612:
609:
606:
600:
597:
575:
571:
562:
546:
542:
535:
532:
529:
523:
503:
494:
491:
484:
479:
454:
450:
447:
444:
399:
395:
392:
389:
377:
375:
365:
358:
355:
352:
346:
340:
335:
324:
321:
318:
308:
305:
300:
296:
289:
287:
279:
276:
273:
270:
267:
264:
261:
258:
255:
250:
245:
242:
239:
235:
231:
229:
222:
218:
204:
197:
187:
177:
168:
159:
150:
141:
132:
127:
120:
118:
116:
111:
102:
100:
96:
91:
86:
82:
76:
70:
65:
61:
57:
53:
49:
45:
36:
26:
21:
8771:Transposable
8635:Narcissistic
8542:Digital root
8462:Super-Poulet
8422:Jordan–Pólya
8371:prime factor
8276:Noncototient
8243:Almost prime
8225:Superperfect
8200:Refactorable
8195:Quasiperfect
8170:Hyperperfect
8011:Pseudoprimes
7982:Wall–Sun–Sun
7917:Ordered Bell
7887:Fuss–Catalan
7799:non-centered
7749:Dodecahedral
7726:non-centered
7619:
7612:non-centered
7514:Wolstenholme
7259:× 2 ± 1
7256:
7255:Of the form
7222:Eighth power
7202:Fourth power
7008:Power series
6769:
6750:Lucas number
6702:Powers of 10
6682:Cubic number
6567:non-centered
6527:non-centered
6462:Cube numbers
6454:non-centered
6352:
6345:non-centered
6335:Star numbers
6232:
6218:
6205:cut-the-knot
6196:cut-the-knot
6178:
6132:
6126:
6118:
6113:
6103:
6050:
6043:
6015:
6008:
5996:. Retrieved
5985:
5973:. Retrieved
5953:
5949:
5939:
5918:
5885:
5881:
5875:
5864:
5853:
5820:
5816:
5806:
5797:
5788:
5782:
5770:. Retrieved
5766:the original
5761:
5752:
5739:
5730:
5718:. Retrieved
5707:
5696:. Retrieved
5692:the original
5687:
5677:
5657:
5647:
5638:
5629:
5590:
5584:
5576:Donald Knuth
5573:
5545:
5530:
5471:
5466:
5462:
5456:
5441:
5360:
5287:
5193:
5107:
5101:
5095:
5089:
5083:
5072:
5068:
5060:
5056:
5048:
5044:
5037:
5030:
5026:
5018:depreciation
5015:
5007:
5003:
4997:
4990:
4982:
4978:
4972:
4965:
4959:
4955:
4951:
4942:Applications
4834:
4770:
4523:
4512:
4498:
4483:
4463:
4460:
4298:
4221:
4087:
4079:
4074:
4066:
4061:
4053:
4048:
4040:
4035:
4028:
4024:
4022:is equal to
4018:
4011:
4003:
3995:
3987:
3979:
3971:
3963:
3956:
3950:
3943:
3938:
3934:
3927:
3925:. Note that
3910:
3902:
3896:
3890:
3886:
3880:
3877:
3874:
3871:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3839:
3836:
3833:
3830:
3824:digital root
3817:
3814:
3811:
3798:
3684:
3681:
3657:
3639:
3620:
3616:
3612:
3608:
3120:
3113:
3107:
3104:
3041:
3039:th centered
3035:
3016:
3008:
3001:
2993:
2986:
2983:
2827:
2820:
2720:
2712:
2707:
2700:
2694:
2540:
2539:
2426:
2415:
2398:
2058:
2055:
1971:
1803:
1797:
1796:
1794:
1704:
1698:
1697:
1695:
1689:
1688:
1683:
1682:
1681:and denoted
1675:Donald Knuth
1670:
1668:
1662:
1657:
1648:
1640:
1636:
1630:
1623:
1618:
1612:
1608:
1605:
1592:
1580:Pythagoreans
1573:
1557:
1546:
1417:
1105:
972:
868:
841:
561:visual proof
495:
489:
482:
426:
202:
148:
106:
89:
80:
74:
68:
64:cube numbers
47:
43:
41:
24:
8792:Extravagant
8787:Equidigital
8742:permutation
8701:Palindromic
8675:Automorphic
8573:Sum-product
8552:Sum-product
8507:Persistence
8402:Erdős–Woods
8324:Untouchable
8205:Semiperfect
8155:Hemiperfect
7816:Tesseractic
7754:Icosahedral
7734:Tetrahedral
7665:Dodecagonal
7366:Recursively
7237:Prime power
7212:Sixth power
7207:Fifth power
7187:Power of 10
7145:Classes of
6875:Conditional
6863:Convergence
6854:Telescoping
6839:Alternating
6755:Pell number
6559:dimensional
5639:Math Is Fun
5444:square root
4993:group stage
4486:2 − 1
4090:reciprocals
3855:45 = 9 × 5
3852:36 = 9 × 4
3685:Every even
3363:which uses
1617:solves the
9053:Categories
9004:Graphemics
8877:Pernicious
8731:Undulating
8706:Pandigital
8680:Trimorphic
8281:Nontotient
8130:Arithmetic
7744:Octahedral
7645:Heptagonal
7635:Pentagonal
7620:Triangular
7461:Sierpiński
7383:Jacobsthal
7182:Power of 3
7177:Power of 2
6900:Convergent
6844:Convergent
5998:2024-04-14
5930:1601.06378
5823:: 89–100.
5772:12 January
5698:2014-04-16
5622:References
5588:? for the
5580:factorials
5460:such that
5216:Max. pips
5195:Board game
4830:digit sums
3829:0 = 9 × 0
2708:Also, the
1699:10 termial
1634:people is
107:(sequence
87:from 1 to
9084:Triangles
8761:Parasitic
8610:Factorion
8537:Digit sum
8529:Digit sum
8347:Fortunate
8334:Primorial
8248:Semiprime
8185:Practical
8150:Descartes
8145:Deficient
8135:Betrothed
7977:Wieferich
7806:Pentatope
7769:pyramidal
7660:Decagonal
7655:Nonagonal
7650:Octagonal
7640:Hexagonal
7499:Practical
7446:Congruent
7378:Fibonacci
7342:Loeschian
6931:Divergent
6849:Divergent
6711:Advanced
6687:Factorial
6635:Sequences
6492:pyramidal
6220:MathWorld
6185:EMS Press
6078:198342061
5910:122221070
5902:1382-4090
5845:125426184
5837:1432-1815
5615:Tetractys
5508:−
5010:− 1
4985:− 1
4694:−
4605:−
4543:−
4438:−
4422:−
4250:∞
4235:∑
4175:∞
4160:∑
4116:∞
4101:∑
3715:−
3574:−
3502:∑
3467:∑
3463:…
3453:−
3429:−
3417:∑
3388:−
3376:∑
3267:∑
3232:∑
3130:∑
3080:−
2878:∑
2844:∑
2773:∑
2733:∑
2597:−
2586:−
2578:−
2367:−
2356:−
2309:−
2209:−
2190:−
2159:−
2089:−
1995:∞
1992:→
1930:−
1910:−
1841:−
1654:factorial
1118:∑
1077:. Adding
1015:∑
890:∑
712:follows:
601:×
274:⋯
236:∑
196:7-simplex
186:6-simplex
176:5-simplex
8833:Friedman
8766:Primeval
8711:Repdigit
8668:-related
8615:Kaprekar
8589:Meertens
8512:Additive
8499:dynamics
8407:Friendly
8319:Sociable
8309:Amicable
8120:Abundant
8100:dynamics
7922:Schröder
7912:Narayana
7882:Eulerian
7872:Delannoy
7867:Dedekind
7688:centered
7554:centered
7441:Amenable
7398:Narayana
7388:Leonardo
7284:Mersenne
7232:Powerful
7172:Achilles
7104:Category
6870:Absolute
6416:centered
6287:centered
6159:53079729
6101:(1810),
5975:25 April
5720:28 March
5658:Calculus
5655:(2008).
5598:See also
5081:, where
5035:, where
3942:+ 1 = (2
2999:and the
1658:products
1588:Computus
95:sequence
9006:related
8970:related
8934:related
8932:Sorting
8817:Vampire
8802:Harshad
8744:related
8716:Repunit
8630:Lychrel
8605:Dudeney
8457:Størmer
8452:Sphenic
8437:Regular
8375:divisor
8314:Perfect
8210:Sublime
8180:Perfect
7907:Motzkin
7862:Catalan
7403:Padovan
7337:Leyland
7332:Idoneal
7327:Hilbert
7299:Woodall
6880:Uniform
6557:Higher
6187:, 2001
5970:3621134
5562:is the
5189:
5177:
5173:
5161:
5157:
5145:
5141:
5129:
5125:
5113:
5078:
5053:
4472:ΕΥΡΗΚΑ!
3968:) are:
3922:
3907:
3820:base 10
1972:In the
1568:
1554:
198:numbers
188:numbers
178:numbers
121:Formula
113:in the
110:A000217
8872:Odious
8797:Frugal
8751:Cyclic
8740:Digit-
8447:Smooth
8432:Pronic
8392:Cyclic
8369:Other
8342:Euclid
7992:Wilson
7966:Primes
7625:Square
7494:Polite
7456:Riesel
7451:Knödel
7413:Perrin
7294:Thabit
7279:Fermat
7269:Cullen
7192:Square
7160:Powers
6832:Series
6639:series
6157:
6147:
6076:
6066:
6031:
5968:
5908:
5900:
5843:
5835:
5665:
5207:domino
3954:given
3822:, the
3796:where
3105:where
3033:; the
1942:
1886:
1883:
1880:
1584:Dicuil
427:where
194:
192:
184:
182:
174:
172:
165:
163:
156:
154:
147:
145:
138:
136:
93:. The
66:. The
8913:Prime
8908:Lucky
8897:sieve
8826:Other
8812:Smith
8692:Digit
8650:Happy
8625:Keith
8598:Other
8442:Rough
8412:Giuga
7877:Euler
7739:Cubic
7393:Lucas
7289:Proth
6775:array
6655:Basic
6155:S2CID
6074:S2CID
5966:JSTOR
5925:arXiv
5906:S2CID
5841:S2CID
5813:(PDF)
5804:"Die
4492:(see
3805:is a
2620:with
2505:with
1974:limit
1673:" by
101:, is
8867:Evil
8547:Self
8497:and
8387:Blum
8098:and
7902:Lobb
7857:Cake
7852:Bell
7602:Star
7509:Ulam
7408:Pell
7197:Cube
6715:list
6637:and
6145:ISBN
6064:ISBN
6029:ISBN
5977:2024
5898:ISSN
5833:ISSN
5774:2022
5722:2015
5663:ISBN
5431:253
5422:190
5419:161
5413:136
5410:120
5051:) ×
4490:4095
4367:and
4014:+ 21
4006:+ 15
3998:+ 10
3946:+ 1)
3019:− 1)
3011:+ 1)
2653:and
115:OEIS
62:and
8985:Ban
8373:or
7892:Lah
6203:at
6194:at
6137:doi
6056:doi
6021:doi
5958:doi
5890:doi
5825:doi
5554:of
5548:+ 1
5446:of
5428:231
5425:210
5416:153
5407:105
5404:91
5401:78
5395:55
5392:45
5386:28
5383:21
5355:22
5346:19
5343:18
5337:16
5334:15
5328:13
5325:12
5319:10
5282:21
5273:18
5270:17
5264:15
5261:14
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5252:11
4970:of
4828:as
4496:).
4488:is
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4069:+ 3
4056:+ 2
4043:+ 1
4010:169
4002:121
3990:+ 6
3982:+ 3
3974:+ 1
3966:+ 0
3913:− 1
3878:If
3869:...
3818:In
3619:th
3611:th
3005:th
2990:th
2831:th
2541:All
1985:lim
1677:'s
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9055::
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6097:;
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6072:.
6062:.
6054:.
6027:.
5964:.
5954:86
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5948:.
5904:.
5896:.
5884:.
5839:.
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4966:A
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2835::
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5467:n
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5371:3
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4706:(
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4467:0
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4441:1
4435:b
4431:T
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4415:T
4411:+
4406:b
4402:T
4396:a
4392:T
4388:=
4383:b
4380:a
4376:T
4355:b
4352:a
4349:+
4344:b
4340:T
4336:+
4331:a
4327:T
4323:=
4318:b
4315:+
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4308:T
4282:=
4276:)
4273:1
4270:+
4267:n
4264:(
4261:n
4257:1
4245:1
4242:=
4239:n
4205:=
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4191:2
4187:n
4182:1
4170:1
4167:=
4164:n
4156:2
4153:=
4146:2
4142:n
4139:+
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4123:1
4111:1
4108:=
4105:n
4080:n
4078:9
4075:T
4067:n
4065:7
4062:T
4054:n
4052:5
4049:T
4041:n
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4025:T
4019:x
4012:x
4004:x
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3988:x
3980:x
3972:x
3970:9
3964:x
3962:1
3957:a
3951:b
3944:n
3939:n
3935:T
3933:8
3928:b
3919:8
3916:/
3911:a
3903:b
3897:a
3891:b
3881:x
3801:p
3799:M
3780:p
3776:M
3771:T
3767:=
3762:2
3758:)
3755:1
3752:+
3747:p
3743:M
3739:(
3734:p
3730:M
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3718:1
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3708:2
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3621:n
3617:k
3613:k
3609:n
3586:)
3581:k
3577:1
3571:k
3568:+
3563:k
3559:n
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3546:=
3541:1
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3529:2
3525:n
3519:1
3516:=
3511:1
3507:n
3494:3
3490:n
3484:1
3481:=
3476:2
3472:n
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3432:2
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3409:k
3405:n
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3333:+
3328:3
3324:n
3317:(
3311:=
3306:1
3302:n
3294:2
3290:n
3284:1
3281:=
3276:1
3272:n
3259:3
3255:n
3249:1
3246:=
3241:2
3237:n
3208:)
3203:2
3199:1
3196:+
3191:2
3187:n
3180:(
3174:=
3169:1
3165:n
3157:2
3153:n
3147:1
3144:=
3139:1
3135:n
3108:T
3091:1
3088:+
3083:1
3077:n
3073:T
3069:k
3066:=
3061:n
3057:k
3053:C
3042:k
3036:n
3017:n
3015:(
3009:m
3007:(
3002:n
2994:m
2987:n
2970:.
2965:6
2961:)
2958:2
2955:+
2952:n
2949:(
2946:)
2943:1
2940:+
2937:n
2934:(
2931:n
2925:=
2920:2
2916:)
2913:1
2910:+
2907:k
2904:(
2901:k
2893:n
2888:1
2885:=
2882:k
2874:=
2869:k
2865:T
2859:n
2854:1
2851:=
2848:k
2828:n
2823:n
2807:.
2802:2
2797:)
2793:k
2788:n
2783:1
2780:=
2777:k
2768:(
2763:=
2758:3
2754:k
2748:n
2743:1
2740:=
2737:k
2721:n
2713:n
2701:n
2695:n
2671:=
2666:1
2662:S
2641:0
2638:=
2633:0
2629:S
2608:2
2605:+
2600:2
2594:n
2590:S
2581:1
2575:n
2571:S
2564:=
2559:n
2555:S
2523:=
2518:1
2514:S
2492:)
2488:1
2485:+
2480:n
2476:S
2472:8
2468:(
2462:n
2458:S
2454:4
2451:=
2446:1
2443:+
2440:n
2436:S
2385:.
2380:2
2376:)
2370:1
2364:n
2360:T
2351:n
2347:T
2343:(
2340:=
2335:2
2331:n
2327:=
2323:)
2317:2
2314:n
2304:2
2299:2
2295:n
2288:(
2284:+
2280:)
2274:2
2271:n
2266:+
2261:2
2256:2
2252:n
2245:(
2241:=
2237:)
2231:2
2221:2
2216:)
2212:1
2206:n
2202:(
2193:1
2187:n
2181:+
2176:2
2171:2
2166:)
2162:1
2156:n
2152:(
2144:(
2140:+
2136:)
2130:2
2127:n
2122:+
2117:2
2112:2
2108:n
2101:(
2097:=
2092:1
2086:n
2082:T
2078:+
2073:n
2069:T
2037:.
2032:3
2029:1
2024:=
2017:n
2013:L
2007:n
2003:T
1989:n
1955:=
1950:1
1946:L
1939:,
1936:)
1933:1
1927:n
1924:(
1921:3
1918:+
1913:1
1907:n
1903:L
1899:=
1894:n
1890:L
1877:;
1871:)
1866:2
1863:n
1858:(
1852:3
1849:=
1844:1
1838:n
1834:T
1830:3
1827:=
1822:n
1818:L
1778:=
1772:+
1769:9
1766:+
1763:8
1760:+
1757:7
1754:+
1751:6
1748:+
1745:5
1742:+
1739:4
1736:+
1733:3
1730:+
1727:2
1724:+
1721:1
1718:=
1715:?
1663:n
1649:T
1641:n
1637:T
1631:n
1624:n
1613:n
1609:T
1574:n
1565:2
1562:/
1558:n
1532:n
1512:3
1492:2
1472:1
1452:1
1449:+
1446:m
1426:m
1400:,
1395:2
1391:)
1388:2
1385:+
1382:m
1379:(
1376:)
1373:1
1370:+
1367:m
1364:(
1358:=
1346:2
1342:2
1339:+
1336:m
1333:3
1330:+
1325:2
1321:m
1314:=
1302:2
1298:2
1295:+
1292:m
1289:2
1286:+
1283:m
1280:+
1275:2
1271:m
1264:=
1252:2
1248:2
1245:+
1242:m
1239:2
1236:+
1233:)
1230:1
1227:+
1224:m
1221:(
1218:m
1212:=
1202:1
1199:+
1196:m
1193:+
1188:2
1184:)
1181:1
1178:+
1175:m
1172:(
1169:m
1163:=
1156:)
1153:1
1150:+
1147:m
1144:(
1141:+
1138:k
1133:m
1128:1
1125:=
1122:k
1091:1
1088:+
1085:m
1063:2
1059:)
1056:1
1053:+
1050:m
1047:(
1044:m
1038:=
1035:k
1030:m
1025:1
1022:=
1019:k
1011:=
1006:m
1002:T
981:m
956:=
951:2
948:2
943:=
938:2
934:)
931:1
928:+
925:1
922:(
919:1
913:=
910:k
905:1
900:1
897:=
894:k
886:=
881:1
877:T
854:1
819:=
814:2
810:)
807:1
804:+
801:4
798:(
795:4
789:=
784:4
780:T
756:=
753:)
750:1
747:+
744:4
741:(
738:4
735:=
730:4
726:T
722:2
698:4
694:T
671:2
667:)
664:1
661:+
658:n
655:(
652:n
646:=
641:n
637:T
616:)
613:1
610:+
607:n
604:(
598:n
576:n
572:T
547:2
543:/
539:)
536:1
533:+
530:n
527:(
524:n
504:n
490:n
483:n
460:)
455:2
451:1
448:+
445:n
439:(
405:)
400:2
396:1
393:+
390:n
384:(
378:=
366:2
362:)
359:1
356:+
353:n
350:(
347:n
341:=
336:2
328:)
325:1
322:+
319:n
316:(
309:n
306:+
301:2
297:n
290:=
280:n
277:+
271:+
268:2
265:+
262:1
259:=
256:k
251:n
246:1
243:=
240:k
232:=
223:n
219:T
90:n
81:n
75:n
69:n
30:)
28:0
25:T
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