882:
2031:
27:
84:
4841:
1774:
2090:. The difference between any two odd perfect squares is a multiple of 8. The difference between 1 and any higher odd perfect square always is eight times a triangular number, while the difference between 9 and any higher odd perfect square is eight times a triangular number minus eight. Since all triangular numbers have an odd factor, but no two values of
2250:
If the number ends in 5, its square will end in 5; similarly for ending in 25, 625, 0625, 90625, ... 8212890625, etc. If the number ends in 6, its square will end in 6, similarly for ending in 76, 376, 9376, 09376, ... 1787109376. For example, the square of 55376 is 3066501376, both ending in
1749:
2360:. An analogous pattern applies for the last 3 digits around multiples of 250, and so on. As a consequence, of the 100 possible last 2 digits, only 22 of them occur among square numbers (since 00 and 25 are repeated).
346:
1434:
is definitely not square. Repeating the divisions of the previous sentence, one concludes that every prime must divide a given perfect square an even number of times (including possibly 0 times). Thus, the number
2021:
1377:
Similar rules can be given for other bases, or for earlier digits (the tens instead of the units digit, for example). All such rules can be proved by checking a fixed number of cases and using
1785:: if a body falling from rest covers one unit of distance in the first arbitrary time interval, it covers 3, 5, 7, etc., units of distance in subsequent time intervals of the same length. From
1571:
872:
379:
2358:
1840:
1259:
1111:
1553:
2943:
409:
249:
2270:
In base 10, the last two digits of square numbers follow a repeating pattern mirrored symmetrical around multiples of 25. In the example of 24 and 26, both 1 off from 25,
1147:
881:
1917:
1042:
1440:
1013:
802:
as can be seen in the above pictures, where a square results from the previous one by adding an odd number of points (shown in magenta). The formula follows:
2578:
2389:
2508:
2264:
1762:
1298:
states that any positive integer can be written as the sum of four or fewer perfect squares. Three squares are not sufficient for numbers of the form
434:
291:
numbers are included, then a square is the ratio of two square integers, and, conversely, the ratio of two square integers is a square, for example,
2936:
1292:
of positive divisors. An integer root is the only divisor that pairs up with itself to yield the square number, while other divisors come in pairs.
659:. Equivalently, it is possible to count square numbers by adding together the last square, the last square's root, and the current root, that is,
294:
2482:
1922:
1769:
0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201...
48:
3743:
2929:
777:
760:
2371:
743:
726:
709:
3738:
1288:
Another property of a square number is that (except 0) it has an odd number of positive divisors, while other natural numbers have an
3753:
3733:
4895:
2541:
2406:
1364:
if a number is divisible both by 2 and by 3 (that is, divisible by 6), its square ends in 0, and its preceding digit must be 0 or 3;
70:
2030:
1781:
The sum of the first odd integers, beginning with one, is a perfect square: 1, 1 + 3, 1 + 3 + 5, 1 + 3 + 5 + 7, etc. This explains
1295:
805:
4446:
4026:
2383:
1782:
187:
points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of
4865:
3748:
2571:
4532:
1191:
1370:
if a number is divisible by 2, but not by 3, its square ends in 4, and its preceding digit must be 0, 1, 4, 5, 8, or 9; and
3848:
1047:
4198:
3517:
3310:
2756:
4233:
4203:
3878:
3868:
1188:
More generally, the difference of the squares of two numbers is the product of their sum and their difference. That is,
4374:
3788:
3522:
3502:
2761:
2741:
4064:
41:
35:
4228:
4875:
4323:
3946:
3703:
3512:
3494:
3388:
3378:
3368:
2751:
2733:
2632:
2622:
2612:
4208:
4870:
4451:
3996:
3617:
3403:
3398:
3393:
3383:
3360:
2856:
2647:
2642:
2637:
2627:
2604:
2564:
2087:
1487:
1282:
1262:
354:
1788:
52:
3436:
2436:
2281:
3693:
4890:
4562:
4527:
4313:
4223:
4097:
4072:
3981:
3971:
3583:
3565:
3485:
2822:
2799:
2724:
1883:
1752:
1744:{\displaystyle \sum _{n=0}^{N}n^{2}=0^{2}+1^{2}+2^{2}+3^{2}+4^{2}+\cdots +N^{2}={\frac {N(N+1)(2N+1)}{6}}.}
1367:
if a number is divisible neither by 2 nor by 3, its square ends in 1, and its preceding digit must be even;
4885:
4822:
4092:
3966:
3597:
3373:
3153:
3080:
2836:
2617:
1373:
if a number is not divisible by 2, but by 3, its square ends in 9, and its preceding digit must be 0 or 6.
1278:
4077:
3931:
3858:
3013:
2891:
1525:
4786:
4426:
1867:, and the distance from the starting point are consecutive squares for integer values of time elapsed.
388:
83:
4719:
4613:
4577:
4318:
4041:
4021:
3838:
3507:
3295:
3267:
2746:
1289:
799:
221:
4441:
4305:
4300:
4268:
4031:
4006:
4001:
3976:
3906:
3902:
3833:
3723:
3555:
3351:
3320:
2789:
2595:
2400:
2034:
1319:
1307:
259:
4840:
1120:
4844:
4598:
4593:
4507:
4481:
4379:
4358:
4130:
4011:
3961:
3883:
3853:
3793:
3560:
3540:
3471:
3184:
2817:
2794:
2710:
2430:
2418:
2256:
2053:
1378:
200:
3728:
2083:
is always even. In other words, all odd square numbers have a remainder of 1 when divided by 8.
4880:
4738:
4683:
4537:
4512:
4486:
4263:
3941:
3936:
3863:
3843:
3828:
3550:
3532:
3451:
3441:
3426:
3204:
3189:
2784:
2771:
2690:
2680:
2670:
2537:
2478:
2424:
1274:
2472:
1345:
if the last digit of a number is 4 or 6, its square ends in an odd digit followed by a 6; and
4774:
4567:
4153:
4125:
4115:
4107:
3991:
3956:
3951:
3918:
3612:
3575:
3466:
3461:
3456:
3446:
3418:
3305:
3257:
3252:
3209:
3148:
2851:
2809:
2705:
2700:
2695:
2685:
2662:
2458:
2395:
112:
1895:
4750:
4639:
4572:
4498:
4421:
4395:
4213:
3926:
3783:
3718:
3688:
3678:
3673:
3339:
3247:
3194:
3038:
2978:
2587:
1342:
if the last digit of a number is 3 or 7, its square ends in an even digit followed by a 9;
1339:
if the last digit of a number is 2 or 8, its square ends in an even digit followed by a 4;
1336:
if the last digit of a number is 1 or 9, its square ends in an even digit followed by a 1;
1018:
288:
192:
88:
992:
4755:
4623:
4608:
4472:
4436:
4411:
4287:
4258:
4243:
4120:
4016:
3986:
3713:
3668:
3545:
3143:
3138:
3133:
3105:
3090:
3003:
2988:
2966:
2953:
2779:
2377:
1875:
1558:
644:
The difference between any perfect square and its predecessor is given by the identity
412:
196:
145:
116:
4859:
4678:
4662:
4603:
4557:
4253:
4238:
4148:
3873:
3300:
3262:
3219:
3100:
3085:
3075:
3033:
3023:
2901:
1447:
2146:
and represents digits before 25. For example, the square of 65 can be calculated by
1450:
of large numbers. Instead of testing for divisibility, test for squarity: for given
4714:
4703:
4618:
4456:
4431:
4348:
4248:
4218:
4193:
4177:
4082:
4049:
3798:
3772:
3683:
3622:
3199:
3095:
3028:
3008:
2983:
2906:
2861:
2412:
1385:
1357:
1150:
211:
2094:
differ by an amount containing an odd factor, the only perfect square of the form
2551:
1306:. A positive integer can be represented as a sum of two squares precisely if its
4673:
4548:
4353:
3817:
3708:
3663:
3658:
3408:
3315:
3214:
3043:
3018:
2993:
2896:
2652:
2526:
2498:
1773:
215:
207:
166:
119:
of some integer with itself. For example, 9 is a square number, since it equals
96:
776:
759:
4810:
4791:
4087:
3698:
1889:
All fourth powers, sixth powers, eighth powers and so on are perfect squares.
742:
725:
708:
284:
2921:
4416:
4343:
4335:
4140:
4054:
3172:
2876:
1498:
divides 9991. This test is deterministic for odd divisors in the range from
909:
287:
one. The concept of square can be extended to some other number systems. If
1329:, a square number can end only with digits 0, 1, 4, 5, 6 or 9, as follows:
2386: – Product of sums of four squares expressed as a sum of four squares
4517:
2530:
1360:
can end only with prime digits or 1), that is, 0, 1, 4 or 9, as follows:
943:
th square number can be calculated from the previous two by doubling the
341:{\displaystyle \textstyle {\frac {4}{9}}=\left({\frac {2}{3}}\right)^{2}}
2471:
Olenick, Richard P.; Apostol, Tom M.; Goodstein, David L. (2008-01-14).
4522:
4181:
1353:
1326:
255:
108:
2374: – Expression of a product of sums of squares as a sum of squares
1356:, a square number can end only with square digits (like in base 12, a
2439: – Integer that is both a perfect square and a triangular number
2503:"Sequence A003226 (Automorphic numbers: n^2 ends with n.)"
2556:
2392: – Condition under which an odd prime is a sum of two squares
2029:
880:
82:
2041:
Squares of even numbers are even, and are divisible by 4, since
162:
4808:
4772:
4736:
4700:
4660:
4285:
4174:
3900:
3815:
3770:
3647:
3337:
3284:
3236:
3170:
3122:
3060:
2964:
2925:
2560:
20:
916:
th square number can be computed from the previous square by
2427: – Integer that is a perfect square modulo some integer
2502:
2474:
The
Mechanical Universe: Introduction to Mechanics and Heat
2260:
1757:
429:
2403: – Greatest integer less than or equal to square root
1348:
if the last digit of a number is 5, its square ends in 25.
1333:
if the last digit of a number is 0, its square ends in 00;
1265:, which can be useful for mental arithmetic: for example,
2052:. Squares of odd numbers are odd, and are congruent to 1
1855:(acceleration due to gravity without air resistance); so
1174:
on the right side of the equation above, it follows that
2016:{\displaystyle (T_{n})^{2}+(T_{n+1})^{2}=T_{(n+1)^{2}}}
912:
methods for computing square numbers. For example, the
2284:
1808:
1777:
Proof without words for the sum of odd numbers theorem
979:
2 × 5 − 4 + 2 = 2 × 25 − 16 + 2 = 50 − 16 + 2 = 36 = 6
298:
1925:
1898:
1791:
1574:
1528:
1273:. A square number is also the sum of two consecutive
1194:
1123:
1050:
1021:
995:
808:
391:
357:
297:
224:
214:. A non-negative integer is a square number when its
161:
number comes from the name of the shape. The unit of
2548:
Amazing
Properties of Squares and Their Calculations
1486:. (This is an application of the factorization of a
4632:
4586:
4546:
4497:
4471:
4404:
4388:
4367:
4334:
4299:
4139:
4106:
4063:
4040:
3917:
3605:
3596:
3574:
3531:
3493:
3484:
3417:
3359:
3350:
2884:
2874:
2844:
2835:
2808:
2770:
2732:
2723:
2661:
2603:
2594:
2536:. New York: Springer-Verlag, pp. 30–32, 1996.
2037:
that all centered octagonal numbers are odd squares
1446:Squarity testing can be used as alternative way in
2352:
2015:
1911:
1834:
1743:
1547:
1253:
1141:
1105:
1036:
1007:
866:
689:is a square number if and only if one can arrange
403:
373:
340:
243:
1277:. The sum of two consecutive square numbers is a
1178:is the only prime number one less than a square (
2421: – Integer side lengths of a right triangle
2189:If the number has two digits and is of the form
2409: – Algorithms for calculating square roots
2169:represents the preceding digits, its square is
2124:represents the preceding digits, its square is
2552:https://books.google.com/books?id=njEtt7rfexEC
2228:. For example, to calculate the square of 57,
2098:is 1, and the only perfect square of the form
1892:A unique relationship with triangular numbers
130:The usual notation for the square of a number
2937:
2572:
2255:. (The numbers 5, 6, 25, 76, etc. are called
1310:contains no odd powers of primes of the form
904:. Animated 3D visualization on a tetrahedron.
795:. This is also equal to the sum of the first
8:
867:{\displaystyle n^{2}=\sum _{k=1}^{n}(2k-1).}
398:
392:
368:
358:
374:{\displaystyle \lfloor {\sqrt {m}}\rfloor }
4805:
4769:
4733:
4697:
4657:
4331:
4296:
4282:
4171:
3914:
3897:
3812:
3767:
3644:
3602:
3490:
3356:
3347:
3334:
3281:
3238:Possessing a specific set of other numbers
3233:
3167:
3119:
3057:
2961:
2944:
2930:
2922:
2881:
2841:
2729:
2600:
2579:
2565:
2557:
2477:. Cambridge University Press. p. 18.
2353:{\textstyle (25n+x)^{2}-(25n-x)^{2}=100nx}
2202:represents the units digit, its square is
1835:{\displaystyle s=ut+{\tfrac {1}{2}}at^{2}}
1439:is a square number if and only if, in its
2509:On-Line Encyclopedia of Integer Sequences
2433: – Polynomial function of degree two
2396:Some identities involving several squares
2332:
2304:
2283:
2005:
1988:
1975:
1959:
1943:
1933:
1924:
1903:
1897:
1826:
1807:
1790:
1693:
1684:
1665:
1652:
1639:
1626:
1613:
1600:
1590:
1579:
1573:
1535:
1527:
1212:
1199:
1193:
1122:
1055:
1049:
1020:
994:
837:
826:
813:
807:
390:
361:
356:
331:
317:
299:
296:
225:
223:
115:of an integer; in other words, it is the
71:Learn how and when to remove this message
2380: – Number raised to the third power
2186:. For example, the square of 70 is 4900.
1772:
959:th square number, and adding 2, because
437:) smaller than 60 = 3600 are:
34:This article includes a list of general
2450:
2390:Fermat's theorem on sums of two squares
183:. If a square number is represented by
2415: – Two raised to an integer power
1254:{\displaystyle a^{2}-b^{2}=(a+b)(a-b)}
254:A positive integer that has no square
2153:which makes the square equal to 4225.
1522:covers some range of natural numbers
191:; thus, square numbers are a type of
7:
1751:The first values of these sums, the
1494:is the square of 3, so consequently
173:). Hence, a square with side length
1106:{\displaystyle m^{2}-1=(m-1)(m+1).}
381:square numbers up to and including
2457:Some authors also call squares of
1548:{\displaystyle k\geq {\sqrt {m}}.}
218:is again an integer. For example,
40:it lacks sufficient corresponding
14:
2407:Methods of computing square roots
2278:, both ending in 76. In general,
1882:first positive integers; this is
1468:is the square of an integer
985:The square minus one of a number
404:{\displaystyle \lfloor x\rfloor }
16:Product of an integer with itself
4839:
4447:Perfect digit-to-digit invariant
1878:is the square of the sum of the
775:
758:
741:
724:
707:
695:
25:
2372:Brahmagupta–Fibonacci identity
2329:
2313:
2301:
2285:
2086:Every odd perfect square is a
2002:
1989:
1972:
1952:
1940:
1926:
1729:
1714:
1711:
1699:
1296:Lagrange's four-square theorem
1248:
1236:
1233:
1221:
1164:is the only non-zero value of
1097:
1085:
1082:
1070:
858:
843:
244:{\displaystyle {\sqrt {9}}=3,}
1:
3286:Expressible via specific sums
2757:Centered dodecahedral numbers
2550:. Kiran Anil Parulekar, 2012
2156:If the number is of the form
2111:If the number is of the form
1392:divides a square number
1281:. Every odd square is also a
1263:difference-of-squares formula
2762:Centered icosahedral numbers
2742:Centered tetrahedral numbers
2384:Euler's four-square identity
1783:Galileo's law of odd numbers
1557:A square number cannot be a
1142:{\displaystyle 6\times 8=48}
165:is defined as the area of a
4375:Multiplicative digital root
2752:Centered octahedral numbers
2633:Centered heptagonal numbers
2623:Centered pentagonal numbers
2613:Centered triangular numbers
2026:Odd and even square numbers
951:th square, subtracting the
351:Starting with 1, there are
265:For a non-negative integer
87:Square number 16 as sum of
4912:
2857:Squared triangular numbers
2648:Centered decagonal numbers
2643:Centered nonagonal numbers
2638:Centered octagonal numbers
2628:Centered hexagonal numbers
2499:Sloane, N. J. A.
1443:, all exponents are even.
1269:can be easily computed as
876:5 = 25 = 1 + 3 + 5 + 7 + 9
4835:
4818:
4804:
4782:
4768:
4746:
4732:
4710:
4696:
4669:
4656:
4452:Perfect digital invariant
4295:
4281:
4189:
4170:
4027:Superior highly composite
3913:
3896:
3824:
3811:
3779:
3766:
3654:
3643:
3346:
3333:
3291:
3280:
3243:
3232:
3180:
3166:
3129:
3118:
3071:
3056:
2974:
2960:
2088:centered octagonal number
1488:difference of two squares
1318:. This is generalized by
1283:centered octagonal number
989:is always the product of
251:so 9 is a square number.
153:, usually pronounced as "
4896:Squares in number theory
4065:Euler's totient function
3849:Euler–Jacobi pseudoprime
3124:Other polynomial numbers
2823:Square pyramidal numbers
2800:Stella octangula numbers
2437:Square triangular number
1753:square pyramidal numbers
1441:canonical representation
1271:50 − 3 = 2500 − 9 = 2491
3879:Somer–Lucas pseudoprime
3869:Lucas–Carmichael number
3704:Lazy caterer's sequence
2618:Centered square numbers
1568:first square numbers is
785:The expression for the
385:, where the expression
55:more precise citations.
3754:Wedderburn–Etherington
3154:Lucky numbers of Euler
2354:
2038:
2017:
1913:
1836:
1778:
1771:
1745:
1595:
1549:
1279:centered square number
1255:
1157:and itself, and since
1143:
1107:
1038:
1009:
905:
868:
842:
427:The squares (sequence
405:
375:
342:
245:
195:(other examples being
123:and can be written as
92:
4866:Elementary arithmetic
4042:Prime omega functions
3859:Frobenius pseudoprime
3649:Combinatorial numbers
3518:Centered dodecahedral
3311:Primary pseudoperfect
2747:Centered cube numbers
2355:
2033:
2018:
1914:
1912:{\displaystyle T_{n}}
1837:
1776:
1767:
1746:
1575:
1550:
1454:and some number
1256:
1144:
1108:
1039:
1010:
939:. Alternatively, the
885:The sum of the first
884:
869:
822:
406:
376:
343:
246:
210:, square numbers are
144:, but the equivalent
86:
4501:-composition related
4301:Arithmetic functions
3903:Arithmetic functions
3839:Elliptic pseudoprime
3523:Centered icosahedral
3503:Centered tetrahedral
2790:Dodecahedral numbers
2282:
2259:. They are sequence
1923:
1896:
1884:Nicomachus's theorem
1789:
1572:
1526:
1192:
1170:to give a factor of
1153:has factors of only
1121:
1048:
1037:{\displaystyle m+1;}
1019:
993:
895:1 + 3 + 5 + ... + (2
806:
789:th square number is
693:points in a square:
389:
355:
295:
273:th square number is
222:
4427:Kaprekar's constant
3947:Colossally abundant
3834:Catalan pseudoprime
3734:Schröder–Hipparchus
3513:Centered octahedral
3389:Centered heptagonal
3379:Centered pentagonal
3369:Centered triangular
2969:and related numbers
2907:8-hypercube numbers
2902:7-hypercube numbers
2897:6-hypercube numbers
2892:5-hypercube numbers
2862:Tesseractic numbers
2818:Tetrahedral numbers
2795:Icosahedral numbers
2711:Dodecagonal numbers
2534:The Book of Numbers
2401:Integer square root
2257:automorphic numbers
2035:Proof without words
1861:is proportional to
1396:then the square of
1308:prime factorization
1113:For example, since
1008:{\displaystyle m-1}
415:of the number
258:except 1 is called
157:squared". The name
134:is not the product
4845:Mathematics portal
4787:Aronson's sequence
4533:Smarandache–Wellin
4290:-dependent numbers
3997:Primitive abundant
3884:Strong pseudoprime
3874:Perrin pseudoprime
3854:Fermat pseudoprime
3794:Wolstenholme prime
3618:Squared triangular
3404:Centered decagonal
3399:Centered nonagonal
3394:Centered octagonal
3384:Centered hexagonal
2785:Octahedral numbers
2691:Heptagonal numbers
2681:Pentagonal numbers
2671:Triangular numbers
2512:. OEIS Foundation.
2431:Quadratic function
2419:Pythagorean triple
2350:
2151:= 6 × (6 + 1) = 42
2039:
2013:
1909:
1832:
1817:
1779:
1741:
1545:
1379:modular arithmetic
1275:triangular numbers
1251:
1139:
1103:
1034:
1005:
908:There are several
906:
864:
401:
371:
338:
337:
241:
208:real number system
201:triangular numbers
93:
4876:Integer sequences
4853:
4852:
4831:
4830:
4800:
4799:
4764:
4763:
4728:
4727:
4692:
4691:
4652:
4651:
4648:
4647:
4467:
4466:
4277:
4276:
4166:
4165:
4162:
4161:
4108:Aliquot sequences
3919:Divisor functions
3892:
3891:
3864:Lucas pseudoprime
3844:Euler pseudoprime
3829:Carmichael number
3807:
3806:
3762:
3761:
3639:
3638:
3635:
3634:
3631:
3630:
3592:
3591:
3480:
3479:
3437:Square triangular
3329:
3328:
3276:
3275:
3228:
3227:
3162:
3161:
3114:
3113:
3052:
3051:
2919:
2918:
2915:
2914:
2870:
2869:
2852:Pentatope numbers
2831:
2830:
2719:
2718:
2706:Decagonal numbers
2701:Nonagonal numbers
2696:Octagonal numbers
2686:Hexagonal numbers
2546:Kiran Parulekar.
2484:978-0-521-71592-8
2425:Quadratic residue
1816:
1755:, are: (sequence
1736:
1540:
1400:must also divide
1384:In general, if a
783:
782:
366:
325:
307:
230:
81:
80:
73:
4903:
4871:Figurate numbers
4843:
4806:
4775:Natural language
4770:
4734:
4702:Generated via a
4698:
4658:
4563:Digit-reassembly
4528:Self-descriptive
4332:
4297:
4283:
4234:Lucas–Carmichael
4224:Harmonic divisor
4172:
4098:Sparsely totient
4073:Highly cototient
3982:Multiply perfect
3972:Highly composite
3915:
3898:
3813:
3768:
3749:Telephone number
3645:
3603:
3584:Square pyramidal
3566:Stella octangula
3491:
3357:
3348:
3340:Figurate numbers
3335:
3282:
3234:
3168:
3120:
3058:
2962:
2946:
2939:
2932:
2923:
2882:
2842:
2730:
2601:
2588:Figurate numbers
2581:
2574:
2567:
2558:
2514:
2513:
2495:
2489:
2488:
2468:
2462:
2461:perfect squares.
2459:rational numbers
2455:
2359:
2357:
2356:
2351:
2337:
2336:
2309:
2308:
2277:
2273:
2246:
2242:
2238:
2234:
2227:
2217:
2207:
2201:
2195:
2185:
2175:
2168:
2162:
2152:
2145:
2130:
2123:
2117:
2101:
2097:
2093:
2082:
2071:
2051:
2022:
2020:
2019:
2014:
2012:
2011:
2010:
2009:
1980:
1979:
1970:
1969:
1948:
1947:
1938:
1937:
1918:
1916:
1915:
1910:
1908:
1907:
1866:
1860:
1854:
1848:
1841:
1839:
1838:
1833:
1831:
1830:
1818:
1809:
1760:
1750:
1748:
1747:
1742:
1737:
1732:
1694:
1689:
1688:
1670:
1669:
1657:
1656:
1644:
1643:
1631:
1630:
1618:
1617:
1605:
1604:
1594:
1589:
1554:
1552:
1551:
1546:
1541:
1536:
1521:
1517:
1507:
1497:
1493:
1490:.) For example,
1485:
1481:
1471:
1467:
1457:
1453:
1438:
1433:
1429:
1428:
1426:
1425:
1420:
1417:
1408:fails to divide
1407:
1403:
1399:
1395:
1391:
1320:Waring's problem
1317:
1305:
1272:
1268:
1260:
1258:
1257:
1252:
1217:
1216:
1204:
1203:
1181:
1177:
1173:
1169:
1163:
1156:
1148:
1146:
1145:
1140:
1116:
1112:
1110:
1109:
1104:
1060:
1059:
1043:
1041:
1040:
1035:
1014:
1012:
1011:
1006:
988:
980:
973:
958:
950:
942:
938:
915:
903:
889:odd integers is
877:
873:
871:
870:
865:
841:
836:
818:
817:
798:
794:
788:
779:
772:
762:
755:
745:
738:
728:
721:
711:
704:
696:
676:
658:
432:
418:
410:
408:
407:
402:
384:
380:
378:
377:
372:
367:
362:
347:
345:
344:
339:
336:
335:
330:
326:
318:
308:
300:
282:
278:
272:
268:
250:
248:
247:
242:
231:
226:
193:figurate numbers
182:
176:
172:
156:
152:
143:
133:
126:
122:
76:
69:
65:
62:
56:
51:this article by
42:inline citations
29:
28:
21:
4911:
4910:
4906:
4905:
4904:
4902:
4901:
4900:
4856:
4855:
4854:
4849:
4827:
4823:Strobogrammatic
4814:
4796:
4778:
4760:
4742:
4724:
4706:
4688:
4665:
4644:
4628:
4587:Divisor-related
4582:
4542:
4493:
4463:
4400:
4384:
4363:
4330:
4303:
4291:
4273:
4185:
4184:related numbers
4158:
4135:
4102:
4093:Perfect totient
4059:
4036:
3967:Highly abundant
3909:
3888:
3820:
3803:
3775:
3758:
3744:Stirling second
3650:
3627:
3588:
3570:
3527:
3476:
3413:
3374:Centered square
3342:
3325:
3287:
3272:
3239:
3224:
3176:
3175:defined numbers
3158:
3125:
3110:
3081:Double Mersenne
3067:
3048:
2970:
2956:
2954:natural numbers
2950:
2920:
2911:
2866:
2827:
2804:
2766:
2715:
2657:
2590:
2585:
2523:
2521:Further reading
2518:
2517:
2497:
2496:
2492:
2485:
2470:
2469:
2465:
2456:
2452:
2447:
2442:
2367:
2328:
2300:
2280:
2279:
2275:
2271:
2244:
2240:
2236:
2229:
2219:
2209:
2203:
2197:
2190:
2177:
2170:
2164:
2157:
2147:
2132:
2125:
2119:
2112:
2108:
2099:
2095:
2091:
2073:
2057:
2042:
2028:
2001:
1984:
1971:
1955:
1939:
1929:
1921:
1920:
1899:
1894:
1893:
1870:The sum of the
1862:
1856:
1850:
1843:
1822:
1787:
1786:
1756:
1695:
1680:
1661:
1648:
1635:
1622:
1609:
1596:
1570:
1569:
1564:The sum of the
1524:
1523:
1519:
1509:
1499:
1495:
1491:
1483:
1473:
1469:
1459:
1455:
1451:
1436:
1431:
1421:
1418:
1413:
1412:
1410:
1409:
1405:
1401:
1397:
1393:
1389:
1311:
1299:
1270:
1266:
1208:
1195:
1190:
1189:
1185:
1179:
1175:
1171:
1165:
1158:
1154:
1119:
1118:
1114:
1051:
1046:
1045:
1017:
1016:
991:
990:
986:
978:
974:. For example,
960:
952:
944:
940:
917:
913:
894:
875:
809:
804:
803:
796:
790:
786:
767:
750:
733:
716:
699:
683:
660:
645:
642:
608:
574:
540:
506:
472:
428:
425:
416:
411:represents the
387:
386:
382:
353:
352:
313:
312:
293:
292:
280:
274:
270:
266:
220:
219:
178:
174:
170:
154:
148:
135:
131:
124:
120:
77:
66:
60:
57:
47:Please help to
46:
30:
26:
17:
12:
11:
5:
4909:
4907:
4899:
4898:
4893:
4891:Quadrilaterals
4888:
4883:
4878:
4873:
4868:
4858:
4857:
4851:
4850:
4848:
4847:
4836:
4833:
4832:
4829:
4828:
4826:
4825:
4819:
4816:
4815:
4809:
4802:
4801:
4798:
4797:
4795:
4794:
4789:
4783:
4780:
4779:
4773:
4766:
4765:
4762:
4761:
4759:
4758:
4756:Sorting number
4753:
4751:Pancake number
4747:
4744:
4743:
4737:
4730:
4729:
4726:
4725:
4723:
4722:
4717:
4711:
4708:
4707:
4701:
4694:
4693:
4690:
4689:
4687:
4686:
4681:
4676:
4670:
4667:
4666:
4663:Binary numbers
4661:
4654:
4653:
4650:
4649:
4646:
4645:
4643:
4642:
4636:
4634:
4630:
4629:
4627:
4626:
4621:
4616:
4611:
4606:
4601:
4596:
4590:
4588:
4584:
4583:
4581:
4580:
4575:
4570:
4565:
4560:
4554:
4552:
4544:
4543:
4541:
4540:
4535:
4530:
4525:
4520:
4515:
4510:
4504:
4502:
4495:
4494:
4492:
4491:
4490:
4489:
4478:
4476:
4473:P-adic numbers
4469:
4468:
4465:
4464:
4462:
4461:
4460:
4459:
4449:
4444:
4439:
4434:
4429:
4424:
4419:
4414:
4408:
4406:
4402:
4401:
4399:
4398:
4392:
4390:
4389:Coding-related
4386:
4385:
4383:
4382:
4377:
4371:
4369:
4365:
4364:
4362:
4361:
4356:
4351:
4346:
4340:
4338:
4329:
4328:
4327:
4326:
4324:Multiplicative
4321:
4310:
4308:
4293:
4292:
4288:Numeral system
4286:
4279:
4278:
4275:
4274:
4272:
4271:
4266:
4261:
4256:
4251:
4246:
4241:
4236:
4231:
4226:
4221:
4216:
4211:
4206:
4201:
4196:
4190:
4187:
4186:
4175:
4168:
4167:
4164:
4163:
4160:
4159:
4157:
4156:
4151:
4145:
4143:
4137:
4136:
4134:
4133:
4128:
4123:
4118:
4112:
4110:
4104:
4103:
4101:
4100:
4095:
4090:
4085:
4080:
4078:Highly totient
4075:
4069:
4067:
4061:
4060:
4058:
4057:
4052:
4046:
4044:
4038:
4037:
4035:
4034:
4029:
4024:
4019:
4014:
4009:
4004:
3999:
3994:
3989:
3984:
3979:
3974:
3969:
3964:
3959:
3954:
3949:
3944:
3939:
3934:
3932:Almost perfect
3929:
3923:
3921:
3911:
3910:
3901:
3894:
3893:
3890:
3889:
3887:
3886:
3881:
3876:
3871:
3866:
3861:
3856:
3851:
3846:
3841:
3836:
3831:
3825:
3822:
3821:
3816:
3809:
3808:
3805:
3804:
3802:
3801:
3796:
3791:
3786:
3780:
3777:
3776:
3771:
3764:
3763:
3760:
3759:
3757:
3756:
3751:
3746:
3741:
3739:Stirling first
3736:
3731:
3726:
3721:
3716:
3711:
3706:
3701:
3696:
3691:
3686:
3681:
3676:
3671:
3666:
3661:
3655:
3652:
3651:
3648:
3641:
3640:
3637:
3636:
3633:
3632:
3629:
3628:
3626:
3625:
3620:
3615:
3609:
3607:
3600:
3594:
3593:
3590:
3589:
3587:
3586:
3580:
3578:
3572:
3571:
3569:
3568:
3563:
3558:
3553:
3548:
3543:
3537:
3535:
3529:
3528:
3526:
3525:
3520:
3515:
3510:
3505:
3499:
3497:
3488:
3482:
3481:
3478:
3477:
3475:
3474:
3469:
3464:
3459:
3454:
3449:
3444:
3439:
3434:
3429:
3423:
3421:
3415:
3414:
3412:
3411:
3406:
3401:
3396:
3391:
3386:
3381:
3376:
3371:
3365:
3363:
3354:
3344:
3343:
3338:
3331:
3330:
3327:
3326:
3324:
3323:
3318:
3313:
3308:
3303:
3298:
3292:
3289:
3288:
3285:
3278:
3277:
3274:
3273:
3271:
3270:
3265:
3260:
3255:
3250:
3244:
3241:
3240:
3237:
3230:
3229:
3226:
3225:
3223:
3222:
3217:
3212:
3207:
3202:
3197:
3192:
3187:
3181:
3178:
3177:
3171:
3164:
3163:
3160:
3159:
3157:
3156:
3151:
3146:
3141:
3136:
3130:
3127:
3126:
3123:
3116:
3115:
3112:
3111:
3109:
3108:
3103:
3098:
3093:
3088:
3083:
3078:
3072:
3069:
3068:
3061:
3054:
3053:
3050:
3049:
3047:
3046:
3041:
3036:
3031:
3026:
3021:
3016:
3011:
3006:
3001:
2996:
2991:
2986:
2981:
2975:
2972:
2971:
2965:
2958:
2957:
2951:
2949:
2948:
2941:
2934:
2926:
2917:
2916:
2913:
2912:
2910:
2909:
2904:
2899:
2894:
2888:
2886:
2879:
2872:
2871:
2868:
2867:
2865:
2864:
2859:
2854:
2848:
2846:
2839:
2833:
2832:
2829:
2828:
2826:
2825:
2820:
2814:
2812:
2806:
2805:
2803:
2802:
2797:
2792:
2787:
2782:
2776:
2774:
2768:
2767:
2765:
2764:
2759:
2754:
2749:
2744:
2738:
2736:
2727:
2721:
2720:
2717:
2716:
2714:
2713:
2708:
2703:
2698:
2693:
2688:
2683:
2678:
2676:Square numbers
2673:
2667:
2665:
2659:
2658:
2656:
2655:
2650:
2645:
2640:
2635:
2630:
2625:
2620:
2615:
2609:
2607:
2598:
2592:
2591:
2586:
2584:
2583:
2576:
2569:
2561:
2555:
2554:
2544:
2522:
2519:
2516:
2515:
2490:
2483:
2463:
2449:
2448:
2446:
2443:
2441:
2440:
2434:
2428:
2422:
2416:
2410:
2404:
2398:
2393:
2387:
2381:
2375:
2368:
2366:
2363:
2362:
2361:
2349:
2346:
2343:
2340:
2335:
2331:
2327:
2324:
2321:
2318:
2315:
2312:
2307:
2303:
2299:
2296:
2293:
2290:
2287:
2268:
2248:
2187:
2154:
2107:
2104:
2027:
2024:
2008:
2004:
2000:
1997:
1994:
1991:
1987:
1983:
1978:
1974:
1968:
1965:
1962:
1958:
1954:
1951:
1946:
1942:
1936:
1932:
1928:
1906:
1902:
1829:
1825:
1821:
1815:
1812:
1806:
1803:
1800:
1797:
1794:
1740:
1735:
1731:
1728:
1725:
1722:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1692:
1687:
1683:
1679:
1676:
1673:
1668:
1664:
1660:
1655:
1651:
1647:
1642:
1638:
1634:
1629:
1625:
1621:
1616:
1612:
1608:
1603:
1599:
1593:
1588:
1585:
1582:
1578:
1559:perfect number
1544:
1539:
1534:
1531:
1375:
1374:
1371:
1368:
1365:
1350:
1349:
1346:
1343:
1340:
1337:
1334:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1215:
1211:
1207:
1202:
1198:
1138:
1135:
1132:
1129:
1126:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1058:
1054:
1033:
1030:
1027:
1024:
1004:
1001:
998:
983:
982:
863:
860:
857:
854:
851:
848:
845:
840:
835:
832:
829:
825:
821:
816:
812:
781:
780:
773:
764:
763:
756:
747:
746:
739:
730:
729:
722:
713:
712:
705:
682:
679:
641:
640:
637:
634:
631:
628:
625:
622:
619:
616:
613:
609:
607:
606:
603:
600:
597:
594:
591:
588:
585:
582:
579:
575:
573:
572:
569:
566:
563:
560:
557:
554:
551:
548:
545:
541:
539:
538:
535:
532:
529:
526:
523:
520:
517:
514:
511:
507:
505:
504:
501:
498:
495:
492:
489:
486:
483:
480:
477:
473:
471:
470:
467:
464:
461:
458:
455:
452:
449:
446:
443:
439:
424:
421:
400:
397:
394:
370:
365:
360:
334:
329:
324:
321:
316:
311:
306:
303:
240:
237:
234:
229:
146:exponentiation
105:perfect square
79:
78:
33:
31:
24:
15:
13:
10:
9:
6:
4:
3:
2:
4908:
4897:
4894:
4892:
4889:
4887:
4886:Number theory
4884:
4882:
4879:
4877:
4874:
4872:
4869:
4867:
4864:
4863:
4861:
4846:
4842:
4838:
4837:
4834:
4824:
4821:
4820:
4817:
4812:
4807:
4803:
4793:
4790:
4788:
4785:
4784:
4781:
4776:
4771:
4767:
4757:
4754:
4752:
4749:
4748:
4745:
4740:
4735:
4731:
4721:
4718:
4716:
4713:
4712:
4709:
4705:
4699:
4695:
4685:
4682:
4680:
4677:
4675:
4672:
4671:
4668:
4664:
4659:
4655:
4641:
4638:
4637:
4635:
4631:
4625:
4622:
4620:
4617:
4615:
4614:Polydivisible
4612:
4610:
4607:
4605:
4602:
4600:
4597:
4595:
4592:
4591:
4589:
4585:
4579:
4576:
4574:
4571:
4569:
4566:
4564:
4561:
4559:
4556:
4555:
4553:
4550:
4545:
4539:
4536:
4534:
4531:
4529:
4526:
4524:
4521:
4519:
4516:
4514:
4511:
4509:
4506:
4505:
4503:
4500:
4496:
4488:
4485:
4484:
4483:
4480:
4479:
4477:
4474:
4470:
4458:
4455:
4454:
4453:
4450:
4448:
4445:
4443:
4440:
4438:
4435:
4433:
4430:
4428:
4425:
4423:
4420:
4418:
4415:
4413:
4410:
4409:
4407:
4403:
4397:
4394:
4393:
4391:
4387:
4381:
4378:
4376:
4373:
4372:
4370:
4368:Digit product
4366:
4360:
4357:
4355:
4352:
4350:
4347:
4345:
4342:
4341:
4339:
4337:
4333:
4325:
4322:
4320:
4317:
4316:
4315:
4312:
4311:
4309:
4307:
4302:
4298:
4294:
4289:
4284:
4280:
4270:
4267:
4265:
4262:
4260:
4257:
4255:
4252:
4250:
4247:
4245:
4242:
4240:
4237:
4235:
4232:
4230:
4227:
4225:
4222:
4220:
4217:
4215:
4212:
4210:
4207:
4205:
4204:Erdős–Nicolas
4202:
4200:
4197:
4195:
4192:
4191:
4188:
4183:
4179:
4173:
4169:
4155:
4152:
4150:
4147:
4146:
4144:
4142:
4138:
4132:
4129:
4127:
4124:
4122:
4119:
4117:
4114:
4113:
4111:
4109:
4105:
4099:
4096:
4094:
4091:
4089:
4086:
4084:
4081:
4079:
4076:
4074:
4071:
4070:
4068:
4066:
4062:
4056:
4053:
4051:
4048:
4047:
4045:
4043:
4039:
4033:
4030:
4028:
4025:
4023:
4022:Superabundant
4020:
4018:
4015:
4013:
4010:
4008:
4005:
4003:
4000:
3998:
3995:
3993:
3990:
3988:
3985:
3983:
3980:
3978:
3975:
3973:
3970:
3968:
3965:
3963:
3960:
3958:
3955:
3953:
3950:
3948:
3945:
3943:
3940:
3938:
3935:
3933:
3930:
3928:
3925:
3924:
3922:
3920:
3916:
3912:
3908:
3904:
3899:
3895:
3885:
3882:
3880:
3877:
3875:
3872:
3870:
3867:
3865:
3862:
3860:
3857:
3855:
3852:
3850:
3847:
3845:
3842:
3840:
3837:
3835:
3832:
3830:
3827:
3826:
3823:
3819:
3814:
3810:
3800:
3797:
3795:
3792:
3790:
3787:
3785:
3782:
3781:
3778:
3774:
3769:
3765:
3755:
3752:
3750:
3747:
3745:
3742:
3740:
3737:
3735:
3732:
3730:
3727:
3725:
3722:
3720:
3717:
3715:
3712:
3710:
3707:
3705:
3702:
3700:
3697:
3695:
3692:
3690:
3687:
3685:
3682:
3680:
3677:
3675:
3672:
3670:
3667:
3665:
3662:
3660:
3657:
3656:
3653:
3646:
3642:
3624:
3621:
3619:
3616:
3614:
3611:
3610:
3608:
3604:
3601:
3599:
3598:4-dimensional
3595:
3585:
3582:
3581:
3579:
3577:
3573:
3567:
3564:
3562:
3559:
3557:
3554:
3552:
3549:
3547:
3544:
3542:
3539:
3538:
3536:
3534:
3530:
3524:
3521:
3519:
3516:
3514:
3511:
3509:
3508:Centered cube
3506:
3504:
3501:
3500:
3498:
3496:
3492:
3489:
3487:
3486:3-dimensional
3483:
3473:
3470:
3468:
3465:
3463:
3460:
3458:
3455:
3453:
3450:
3448:
3445:
3443:
3440:
3438:
3435:
3433:
3430:
3428:
3425:
3424:
3422:
3420:
3416:
3410:
3407:
3405:
3402:
3400:
3397:
3395:
3392:
3390:
3387:
3385:
3382:
3380:
3377:
3375:
3372:
3370:
3367:
3366:
3364:
3362:
3358:
3355:
3353:
3352:2-dimensional
3349:
3345:
3341:
3336:
3332:
3322:
3319:
3317:
3314:
3312:
3309:
3307:
3304:
3302:
3299:
3297:
3296:Nonhypotenuse
3294:
3293:
3290:
3283:
3279:
3269:
3266:
3264:
3261:
3259:
3256:
3254:
3251:
3249:
3246:
3245:
3242:
3235:
3231:
3221:
3218:
3216:
3213:
3211:
3208:
3206:
3203:
3201:
3198:
3196:
3193:
3191:
3188:
3186:
3183:
3182:
3179:
3174:
3169:
3165:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3137:
3135:
3132:
3131:
3128:
3121:
3117:
3107:
3104:
3102:
3099:
3097:
3094:
3092:
3089:
3087:
3084:
3082:
3079:
3077:
3074:
3073:
3070:
3065:
3059:
3055:
3045:
3042:
3040:
3037:
3035:
3034:Perfect power
3032:
3030:
3027:
3025:
3024:Seventh power
3022:
3020:
3017:
3015:
3012:
3010:
3007:
3005:
3002:
3000:
2997:
2995:
2992:
2990:
2987:
2985:
2982:
2980:
2977:
2976:
2973:
2968:
2963:
2959:
2955:
2947:
2942:
2940:
2935:
2933:
2928:
2927:
2924:
2908:
2905:
2903:
2900:
2898:
2895:
2893:
2890:
2889:
2887:
2883:
2880:
2878:
2873:
2863:
2860:
2858:
2855:
2853:
2850:
2849:
2847:
2843:
2840:
2838:
2837:4-dimensional
2834:
2824:
2821:
2819:
2816:
2815:
2813:
2811:
2807:
2801:
2798:
2796:
2793:
2791:
2788:
2786:
2783:
2781:
2778:
2777:
2775:
2773:
2769:
2763:
2760:
2758:
2755:
2753:
2750:
2748:
2745:
2743:
2740:
2739:
2737:
2735:
2731:
2728:
2726:
2725:3-dimensional
2722:
2712:
2709:
2707:
2704:
2702:
2699:
2697:
2694:
2692:
2689:
2687:
2684:
2682:
2679:
2677:
2674:
2672:
2669:
2668:
2666:
2664:
2660:
2654:
2651:
2649:
2646:
2644:
2641:
2639:
2636:
2634:
2631:
2629:
2626:
2624:
2621:
2619:
2616:
2614:
2611:
2610:
2608:
2606:
2602:
2599:
2597:
2596:2-dimensional
2593:
2589:
2582:
2577:
2575:
2570:
2568:
2563:
2562:
2559:
2553:
2549:
2545:
2543:
2542:0-387-97993-X
2539:
2535:
2532:
2528:
2527:Conway, J. H.
2525:
2524:
2520:
2511:
2510:
2504:
2500:
2494:
2491:
2486:
2480:
2476:
2475:
2467:
2464:
2460:
2454:
2451:
2444:
2438:
2435:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2411:
2408:
2405:
2402:
2399:
2397:
2394:
2391:
2388:
2385:
2382:
2379:
2376:
2373:
2370:
2369:
2364:
2347:
2344:
2341:
2338:
2333:
2325:
2322:
2319:
2316:
2310:
2305:
2297:
2294:
2291:
2288:
2269:
2266:
2262:
2258:
2254:
2249:
2232:
2226:
2222:
2216:
2212:
2206:
2200:
2194:
2188:
2184:
2180:
2173:
2167:
2160:
2155:
2150:
2143:
2139:
2135:
2128:
2122:
2115:
2110:
2109:
2106:Special cases
2105:
2103:
2089:
2084:
2080:
2076:
2069:
2065:
2061:
2055:
2050:
2046:
2036:
2032:
2025:
2023:
2006:
1998:
1995:
1992:
1985:
1981:
1976:
1966:
1963:
1960:
1956:
1949:
1944:
1934:
1930:
1904:
1900:
1890:
1887:
1885:
1881:
1877:
1873:
1868:
1865:
1859:
1853:
1849:and constant
1846:
1827:
1823:
1819:
1813:
1810:
1804:
1801:
1798:
1795:
1792:
1784:
1775:
1770:
1766:
1764:
1759:
1754:
1738:
1733:
1726:
1723:
1720:
1717:
1708:
1705:
1702:
1696:
1690:
1685:
1681:
1677:
1674:
1671:
1666:
1662:
1658:
1653:
1649:
1645:
1640:
1636:
1632:
1627:
1623:
1619:
1614:
1610:
1606:
1601:
1597:
1591:
1586:
1583:
1580:
1576:
1567:
1562:
1560:
1555:
1542:
1537:
1532:
1529:
1516:
1512:
1506:
1502:
1489:
1480:
1476:
1466:
1462:
1449:
1448:factorization
1444:
1442:
1424:
1416:
1387:
1382:
1380:
1372:
1369:
1366:
1363:
1362:
1361:
1359:
1355:
1347:
1344:
1341:
1338:
1335:
1332:
1331:
1330:
1328:
1323:
1321:
1315:
1309:
1303:
1297:
1293:
1291:
1286:
1284:
1280:
1276:
1264:
1245:
1242:
1239:
1230:
1227:
1224:
1218:
1213:
1209:
1205:
1200:
1196:
1186:
1183:
1168:
1161:
1152:
1136:
1133:
1130:
1127:
1124:
1100:
1094:
1091:
1088:
1079:
1076:
1073:
1067:
1064:
1061:
1056:
1052:
1031:
1028:
1025:
1022:
1002:
999:
996:
977:
976:
975:
971:
967:
963:
956:
948:
936:
932:
928:
924:
920:
911:
902:
898:
892:
888:
883:
879:
874:For example,
861:
855:
852:
849:
846:
838:
833:
830:
827:
823:
819:
814:
810:
801:
793:
778:
774:
770:
766:
765:
761:
757:
753:
749:
748:
744:
740:
736:
732:
731:
727:
723:
719:
715:
714:
710:
706:
702:
698:
697:
694:
692:
688:
680:
678:
675:
671:
667:
663:
656:
652:
648:
638:
635:
632:
629:
626:
623:
620:
617:
614:
611:
610:
604:
601:
598:
595:
592:
589:
586:
583:
580:
577:
576:
570:
567:
564:
561:
558:
555:
552:
549:
546:
543:
542:
536:
533:
530:
527:
524:
521:
518:
515:
512:
509:
508:
502:
499:
496:
493:
490:
487:
484:
481:
478:
475:
474:
468:
465:
462:
459:
456:
453:
450:
447:
444:
441:
440:
438:
436:
431:
422:
420:
414:
395:
363:
349:
332:
327:
322:
319:
314:
309:
304:
301:
290:
286:
277:
263:
261:
257:
252:
238:
235:
232:
227:
217:
213:
209:
204:
202:
198:
194:
190:
186:
181:
168:
164:
160:
151:
147:
142:
138:
128:
118:
114:
110:
106:
102:
101:square number
98:
90:
85:
75:
72:
64:
61:February 2012
54:
50:
44:
43:
37:
32:
23:
22:
19:
4578:Transposable
4442:Narcissistic
4349:Digital root
4269:Super-Poulet
4229:Jordan–Pólya
4178:prime factor
4083:Noncototient
4050:Almost prime
4032:Superperfect
4007:Refactorable
4002:Quasiperfect
3977:Hyperperfect
3818:Pseudoprimes
3789:Wall–Sun–Sun
3724:Ordered Bell
3694:Fuss–Catalan
3606:non-centered
3556:Dodecahedral
3533:non-centered
3431:
3419:non-centered
3321:Wolstenholme
3066:× 2 ± 1
3063:
3062:Of the form
3029:Eighth power
3009:Fourth power
2998:
2885:non-centered
2845:non-centered
2780:Cube numbers
2772:non-centered
2675:
2663:non-centered
2653:Star numbers
2547:
2533:
2506:
2493:
2473:
2466:
2453:
2413:Power of two
2378:Cubic number
2252:
2230:
2224:
2220:
2214:
2210:
2204:
2198:
2192:
2182:
2178:
2171:
2165:
2158:
2148:
2141:
2137:
2133:
2126:
2120:
2113:
2085:
2078:
2074:
2067:
2063:
2059:
2048:
2044:
2040:
1891:
1888:
1879:
1871:
1869:
1863:
1857:
1851:
1844:
1780:
1768:
1565:
1563:
1556:
1514:
1510:
1504:
1500:
1478:
1474:
1464:
1460:
1445:
1422:
1414:
1383:
1376:
1358:prime number
1351:
1324:
1313:
1301:
1294:
1287:
1261:This is the
1187:
1184:
1166:
1159:
1151:prime number
984:
969:
965:
961:
954:
946:
934:
930:
929:− 1) + n = (
926:
922:
918:
907:
900:
896:
890:
886:
791:
784:
768:
751:
734:
717:
700:
690:
686:
684:
673:
669:
665:
661:
654:
650:
646:
643:
426:
350:
275:
264:
253:
212:non-negative
205:
197:cube numbers
188:
184:
179:
158:
149:
140:
136:
129:
111:that is the
104:
100:
94:
67:
58:
39:
18:
4599:Extravagant
4594:Equidigital
4549:permutation
4508:Palindromic
4482:Automorphic
4380:Sum-product
4359:Sum-product
4314:Persistence
4209:Erdős–Woods
4131:Untouchable
4012:Semiperfect
3962:Hemiperfect
3623:Tesseractic
3561:Icosahedral
3541:Tetrahedral
3472:Dodecagonal
3173:Recursively
3044:Prime power
3019:Sixth power
3014:Fifth power
2994:Power of 10
2952:Classes of
2877:dimensional
2237:25 + 7 = 32
1290:even number
800:odd numbers
685:The number
260:square-free
216:square root
167:unit square
97:mathematics
53:introducing
4860:Categories
4811:Graphemics
4684:Pernicious
4538:Undulating
4513:Pandigital
4487:Trimorphic
4088:Nontotient
3937:Arithmetic
3551:Octahedral
3452:Heptagonal
3442:Pentagonal
3427:Triangular
3268:Sierpiński
3190:Jacobsthal
2989:Power of 3
2984:Power of 2
2531:Guy, R. K.
1492:100 − 9991
1149:. Since a
1117:, one has
681:Properties
283:being the
36:references
4568:Parasitic
4417:Factorion
4344:Digit sum
4336:Digit sum
4154:Fortunate
4141:Primorial
4055:Semiprime
3992:Practical
3957:Descartes
3952:Deficient
3942:Betrothed
3784:Wieferich
3613:Pentatope
3576:pyramidal
3467:Decagonal
3462:Nonagonal
3457:Octagonal
3447:Hexagonal
3306:Practical
3253:Congruent
3185:Fibonacci
3149:Loeschian
2810:pyramidal
2323:−
2311:−
2245:57 = 3249
2056:8, since
1675:⋯
1577:∑
1533:≥
1243:−
1206:−
1180:3 = 2 − 1
1128:×
1077:−
1062:−
1000:−
933:− 1) + (2
910:recursive
853:−
824:∑
639:59 = 3481
636:58 = 3364
633:57 = 3249
630:56 = 3136
627:55 = 3025
624:54 = 2916
621:53 = 2809
618:52 = 2704
615:51 = 2601
612:50 = 2500
605:49 = 2401
602:48 = 2304
599:47 = 2209
596:46 = 2116
593:45 = 2025
590:44 = 1936
587:43 = 1849
584:42 = 1764
581:41 = 1681
578:40 = 1600
571:39 = 1521
568:38 = 1444
565:37 = 1369
562:36 = 1296
559:35 = 1225
556:34 = 1156
553:33 = 1089
550:32 = 1024
399:⌋
393:⌊
369:⌋
359:⌊
177:has area
4881:Integers
4640:Friedman
4573:Primeval
4518:Repdigit
4475:-related
4422:Kaprekar
4396:Meertens
4319:Additive
4306:dynamics
4214:Friendly
4126:Sociable
4116:Amicable
3927:Abundant
3907:dynamics
3729:Schröder
3719:Narayana
3689:Eulerian
3679:Delannoy
3674:Dedekind
3495:centered
3361:centered
3248:Amenable
3205:Narayana
3195:Leonardo
3091:Mersenne
3039:Powerful
2979:Achilles
2734:centered
2605:centered
2365:See also
2276:26 = 676
2272:24 = 576
2070:+ 1) + 1
2062:+ 1) = 4
1482:divides
1044:that is,
972:− 2) + 2
968:− 1) − (
925:− 1) + (
771:= 5 = 25
754:= 4 = 16
668:− 1) + (
653:− 1) = 2
547:31 = 961
544:30 = 900
537:29 = 841
534:28 = 784
531:27 = 729
528:26 = 676
525:25 = 625
522:24 = 576
519:23 = 529
516:22 = 484
513:21 = 441
510:20 = 400
503:19 = 361
500:18 = 324
497:17 = 289
494:16 = 256
491:15 = 225
488:14 = 196
485:13 = 169
482:12 = 144
479:11 = 121
476:10 = 100
423:Examples
289:rational
256:divisors
4813:related
4777:related
4741:related
4739:Sorting
4624:Vampire
4609:Harshad
4551:related
4523:Repunit
4437:Lychrel
4412:Dudeney
4264:Størmer
4259:Sphenic
4244:Regular
4182:divisor
4121:Perfect
4017:Sublime
3987:Perfect
3714:Motzkin
3669:Catalan
3210:Padovan
3144:Leyland
3139:Idoneal
3134:Hilbert
3106:Woodall
2875:Higher
2501:(ed.).
2263:in the
2261:A003226
2213:= 25 +
1761:in the
1758:A000330
1496:100 − 3
1430:, then
1427:
1411:
1354:base 12
1327:base 10
1267:47 × 53
899:− 1) =
737:= 3 = 9
720:= 2 = 4
703:= 1 = 1
672:− 1) +
433:in the
430:A000290
279:, with
206:In the
117:product
109:integer
89:gnomons
49:improve
4679:Odious
4604:Frugal
4558:Cyclic
4547:Digit-
4254:Smooth
4239:Pronic
4199:Cyclic
4176:Other
4149:Euclid
3799:Wilson
3773:Primes
3432:Square
3301:Polite
3263:Riesel
3258:Knödel
3220:Perrin
3101:Thabit
3086:Fermat
3076:Cullen
2999:Square
2967:Powers
2540:
2481:
2241:7 = 49
2208:where
2196:where
2176:where
2163:where
2131:where
2118:where
2102:is 9.
2072:, and
2054:modulo
1874:first
1842:, for
1518:where
1388:
1115:7 = 49
469:9 = 81
466:8 = 64
463:7 = 49
460:6 = 36
457:5 = 25
454:4 = 16
285:zeroth
269:, the
159:square
113:square
107:is an
38:, but
4720:Prime
4715:Lucky
4704:sieve
4633:Other
4619:Smith
4499:Digit
4457:Happy
4432:Keith
4405:Other
4249:Rough
4219:Giuga
3684:Euler
3546:Cubic
3200:Lucas
3096:Proth
2445:Notes
2243:, so
2100:2 + 1
2096:2 − 1
2047:) = 4
1876:cubes
1472:then
1458:, if
1404:; if
1386:prime
451:3 = 9
448:2 = 4
445:1 = 1
442:0 = 0
413:floor
281:0 = 0
171:1 × 1
125:3 × 3
4674:Evil
4354:Self
4304:and
4194:Blum
3905:and
3709:Lobb
3664:Cake
3659:Bell
3409:Star
3316:Ulam
3215:Pell
3004:Cube
2538:ISBN
2529:and
2507:The
2479:ISBN
2274:and
2265:OEIS
2239:and
2235:and
2218:and
2205:aabb
2144:+ 1)
2081:+ 1)
1763:OEIS
1304:+ 7)
1015:and
964:= 2(
957:− 2)
949:− 1)
937:− 1)
435:OEIS
199:and
163:area
99:, a
4792:Ban
4180:or
3699:Lah
2342:100
2253:376
2233:= 7
1919:is:
1847:= 0
1508:to
1352:In
1325:In
1316:+ 3
1300:4(8
1182:).
1162:= 2
921:= (
664:= (
657:− 1
649:− (
203:).
103:or
95:In
4862::
2505:.
2317:25
2289:25
2267:.)
2223:=
2221:bb
2211:aa
2181:=
2174:00
2136:=
2129:25
2058:(2
2043:(2
1886:.
1765:)
1561:.
1513:+
1503:−
1477:−
1463:−
1381:.
1322:.
1285:.
1137:48
893:.
878:.
677:.
419:.
348:.
262:.
139:×
127:.
3064:a
2945:e
2938:t
2931:v
2580:e
2573:t
2566:v
2487:.
2348:x
2345:n
2339:=
2334:2
2330:)
2326:x
2320:n
2314:(
2306:2
2302:)
2298:x
2295:+
2292:n
2286:(
2247:.
2231:m
2225:m
2215:m
2199:m
2193:m
2191:5
2183:m
2179:n
2172:n
2166:m
2161:0
2159:m
2149:n
2142:m
2140:(
2138:m
2134:n
2127:n
2121:m
2116:5
2114:m
2092:2
2079:n
2077:(
2075:n
2068:n
2066:(
2064:n
2060:n
2049:n
2045:n
2007:2
2003:)
1999:1
1996:+
1993:n
1990:(
1986:T
1982:=
1977:2
1973:)
1967:1
1964:+
1961:n
1957:T
1953:(
1950:+
1945:2
1941:)
1935:n
1931:T
1927:(
1905:n
1901:T
1880:n
1872:n
1864:t
1858:s
1852:a
1845:u
1828:2
1824:t
1820:a
1814:2
1811:1
1805:+
1802:t
1799:u
1796:=
1793:s
1739:.
1734:6
1730:)
1727:1
1724:+
1721:N
1718:2
1715:(
1712:)
1709:1
1706:+
1703:N
1700:(
1697:N
1691:=
1686:2
1682:N
1678:+
1672:+
1667:2
1663:4
1659:+
1654:2
1650:3
1646:+
1641:2
1637:2
1633:+
1628:2
1624:1
1620:+
1615:2
1611:0
1607:=
1602:2
1598:n
1592:N
1587:0
1584:=
1581:n
1566:n
1543:.
1538:m
1530:k
1520:k
1515:n
1511:k
1505:n
1501:k
1484:m
1479:n
1475:k
1470:n
1465:m
1461:k
1456:k
1452:m
1437:m
1432:m
1423:p
1419:/
1415:m
1406:p
1402:m
1398:p
1394:m
1390:p
1314:k
1312:4
1302:m
1249:)
1246:b
1240:a
1237:(
1234:)
1231:b
1228:+
1225:a
1222:(
1219:=
1214:2
1210:b
1201:2
1197:a
1176:3
1172:1
1167:m
1160:m
1155:1
1134:=
1131:8
1125:6
1101:.
1098:)
1095:1
1092:+
1089:m
1086:(
1083:)
1080:1
1074:m
1071:(
1068:=
1065:1
1057:2
1053:m
1032:;
1029:1
1026:+
1023:m
1003:1
997:m
987:m
981:.
970:n
966:n
962:n
955:n
953:(
947:n
945:(
941:n
935:n
931:n
927:n
923:n
919:n
914:n
901:n
897:n
891:n
887:n
862:.
859:)
856:1
850:k
847:2
844:(
839:n
834:1
831:=
828:k
820:=
815:2
811:n
797:n
792:n
787:n
769:m
752:m
735:m
718:m
701:m
691:m
687:m
674:n
670:n
666:n
662:n
655:n
651:n
647:n
417:x
396:x
383:m
364:m
333:2
328:)
323:3
320:2
315:(
310:=
305:9
302:4
276:n
271:n
267:n
239:,
236:3
233:=
228:9
189:n
185:n
180:n
175:n
169:(
155:n
150:n
141:n
137:n
132:n
121:3
91:.
74:)
68:(
63:)
59:(
45:.
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