| ||||
---|---|---|---|---|
Cardinal | sixty-three | |||
Ordinal | 63rd (sixty-third) | |||
Factorization | 32 × 7 | |||
Divisors | 1, 3, 7, 9, 21, 63 | |||
Greek numeral | ΞΓ´ | |||
Roman numeral | LXIII | |||
Binary | 1111112 | |||
Ternary | 21003 | |||
Senary | 1436 | |||
Octal | 778 | |||
Duodecimal | 5312 | |||
Hexadecimal | 3F16 |
63 (sixty-three) is the natural number following 62 and preceding 64.
63 is the sum of the first six powers of 2 (20 + 21 + ... 25). It is the eighth highly cototient number, [1] and the fourth centered octahedral number after 7 and 25. [2] For five unlabeled elements, there are 63 posets. [3]
Sixty-three is the seventh square-prime of the form and the second of the form . It contains a prime aliquot sum of 41, the thirteenth indexed prime; and part of the aliquot sequence (63, 41, 1, 0) within the 41-aliquot tree.
Zsigmondy's theorem states that where are coprime integers for any integer , there exists a primitive prime divisor that divides and does not divide for any positive integer , except for when
and for a special case where with and , which yields . [4]
63 is a Mersenne number of the form with an of , [5] however this does not yield a Mersenne prime, as 63 is the forty-fourth composite number. [6] It is the only number in the Mersenne sequence whose prime factors are each factors of at least one previous element of the sequence (3 and 7, respectively the first and second Mersenne primes). [7] In the list of Mersenne numbers, 63 lies between Mersenne primes 31 and 127, with 127 the thirty-first prime number. [5] The thirty-first odd number, of the simplest form , is 63. [8] It is also the fourth Woodall number of the form with , with the previous members being 1, 7 and 23 (they add to 31, the third Mersenne prime). [9]
In the integer positive definite quadratic matrix representative of all (even and odd) integers, [10] [11] the sum of all nine terms is equal to 63.
63 is the third Delannoy number, which represents the number of pathways in a grid from a southwest corner to a northeast corner, using only single steps northward, eastward, or northeasterly. [12]
63 holds thirty-six integers that are relatively prime with itself (and up to), equivalently its Euler totient. [13] In the classification of finite simple groups of Lie type, 63 and 36 are both exponents that figure in the orders of three exceptional groups of Lie type. The orders of these groups are equivalent to the product between the quotient of (with prime and a positive integer) by the GCD of , and a (in capital pi notation, product over a set of terms): [14]
Lie algebra holds thirty-six positive roots in sixth-dimensional space, while holds sixty-three positive root vectors in the seven-dimensional space (with one hundred and twenty-six total root vectors, twice 63). [15] The thirty-sixth-largest of thirty-seven total complex reflection groups is , with order where the previous has order ; these are associated, respectively, with and [16]
There are 63 uniform polytopes in the sixth dimension that are generated from the abstract hypercubic Coxeter group (sometimes, the demicube is also included in this family), [17] that is associated with classical Chevalley Lie algebra via the orthogonal group and its corresponding special orthogonal Lie algebra (by symmetries shared between unordered and ordered Dynkin diagrams). There are also 36 uniform 6-polytopes that are generated from the simplex Coxeter group, when counting self-dual configurations of the regular 6-simplex separately. [17] In similar fashion, is associated with classical Chevalley Lie algebra through the special linear group and its corresponding special linear Lie algebra.
In the third dimension, there are a total of sixty-three stellations generated with icosahedral symmetry , using Miller's rules; fifty-nine of these are generated by the regular icosahedron and four by the regular dodecahedron, inclusive (as zeroth indexed stellations for regular figures). [18] Though the regular tetrahedron and cube do not produce any stellations, the only stellation of the regular octahedron as a stella octangula is a compound of two self-dual tetrahedra that facets the cube, since it shares its vertex arrangement. Overall, of order 120 contains a total of thirty-one axes of symmetry; [19] specifically, the lattice that is associated with exceptional Lie algebra contains symmetries that can be traced back to the regular icosahedron via the icosians. [20] The icosahedron and dodecahedron can inscribe any of the other three Platonic solids, which are all collectively responsible for generating a maximum of thirty-six polyhedra which are either regular (Platonic), semi-regular (Archimedean), or duals to semi-regular polyhedra containing regular vertex-figures (Catalan), when including four enantiomorphs from two semi-regular snub polyhedra and their duals as well as self-dual forms of the tetrahedron. [21]
Otherwise, the sum of the divisors of sixty-three, , [22] is equal to the constant term that belongs to the principal modular function (McKay–Thompson series) of sporadic group , the second largest such group after the Friendly Giant . [23] This value is also the value of the minimal faithful dimensional representation of the Tits group , [24] the only finite simple group that can categorize as being non-strict of Lie type, or loosely sporadic; that is also twice the faithful dimensional representation of exceptional Lie algebra , in 52 dimensions.
Sixty-three is also:
2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number.
10 (ten) is the even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, the most common system of denoting numbers in both spoken and written language.
12 (twelve) is the natural number following 11 and preceding 13. Twelve is a superior highly composite number, divisible by the numbers from 1 to 4, and 6.
11 (eleven) is the natural number following 10 and preceding 12. It is the first repdigit. In English, it is the smallest positive integer whose name has three syllables.
17 (seventeen) is the natural number following 16 and preceding 18. It is a prime number.
19 (nineteen) is the natural number following 18 and preceding 20. It is a prime number.
90 (ninety) is the natural number following 89 and preceding 91.
23 (twenty-three) is the natural number following 22 and preceding 24.
27 is the natural number following 26 and preceding 28.
72 (seventy-two) is the natural number following 71 and preceding 73. It is half a gross or six dozen.
84 (eighty-four) is the natural number following 83 and preceding 85. It is seven dozens.
32 (thirty-two) is the natural number following 31 and preceding 33.
58 (fifty-eight) is the natural number following 57 and preceding 59.
61 (sixty-one) is the natural number following 60 and preceding 62.
104 is the natural number following 103 and preceding 105.
1728 is the natural number following 1727 and preceding 1729. It is a dozen gross, or one great gross. It is also the number of cubic inches in a cubic foot.
240 is the natural number following 239 and preceding 241.
5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number.
744 is the natural number following 743 and preceding 745.
14 (fourteen) is the natural number following 13 and preceding 15.
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