63 (number)
[3] Sixty-three is the seventh square-prime 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.
,[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
, 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.
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
(in capital pi notation, product over a set of
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
[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).
simplex Coxeter group, when counting self-dual configurations of the regular 6-simplex separately.
is associated with classical Chevalley Lie algebra
In the third dimension, there are a total of sixty-three stellations generated with icosahedral symmetry
[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.
of order 120 contains a total of thirty-one axes of symmetry;[19] specifically, the
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.
that belongs to the principal modular function (McKay–Thompson series)
[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
