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58 (fifty-eight) is the natural number following 57 and preceding 59.
Form
Fifty-eight is the seventeenth semiprime[1] and the ninth with 2 as the lowest non-unitary divisor; thus of the form , where is a higher prime (29).
Number-theoretical
58 is equal to the sum of the first seven consecutive prime numbers:[2]
This is a difference of 1 from the seventeenth prime number and seventh super-prime, 59.[3][4] 58 has an aliquot sum of 32[5] within an aliquot sequence of two composite numbers (58, 32, 13, 1, 0) in the 13-aliquot tree.[6] There is no solution to the equation , making fifty-eight the sixth noncototient;[7] however, the totient summatory function over the first thirteen integers is 58.[8][lower-alpha 1]
On the other hand, the Euler totient of 58 is the second perfect number (28),[10] where the sum-of-divisors of 58 is the third unitary perfect number (90).
58 is also the second non-trivial 11-gonal number, after 30.[11]
Sequence of biprimes
58 is the second member of the fifth cluster of two semiprimes or biprimes (57, 58), following (25, 26) and preceding (118, 119).[12]
More specifically, 58 is the eleventh member in the sequence of consecutive discrete semiprimes that begins,[13]
- (14, 15),[lower-alpha 2]
- (21, 22),[lower-alpha 3]
- (33, 34, 35),[lower-alpha 4]
- (38, 39)
- (57, 58)
58 represents twice the sum between the first two discrete biprimes 14 + 15 = 29, with the first two members of the first such triplet 33 and 34 (or twice 17, the fourth super-prime) respectively the twenty-first and twenty-second composite numbers,[14] and 22 itself the thirteenth composite.[14] (Where also, 58 is the sum of all primes between 2 and 17.) The first triplet is the only triplet in the sequence of consecutive discrete biprimes whose members collectively have prime factorizations that nearly span a set of consecutive prime numbers.
Decimal properties
The fifth repdigit is the product between the thirteenth and fifty-eighth primes,
58 is also the smallest integer in decimal whose square root has a continued fraction with period 7.[16] It is the fourth Smith number whose sum of its digits is equal to the sum of the digits in its prime factorization (13).[17]
Mertens function
Given 58, the Mertens function returns , the fourth such number to do so.[18] The sum of the first three numbers to return zero (2, 39, 40) sum to 81 = 92, which is the fifty-eighth composite number.[14]
Geometric properties
The regular icosahedron produces fifty-eight distinct stellations, the most of any other Platonic solid, which collectively produce sixty-two stellations.[19][20]
Coxeter groups
With regard to Coxeter groups and uniform polytopes in higher dimensional spaces, there are:
- 58 distinct uniform polytopes in the fifth dimension that are generated from symmetries of three Coxeter groups, they are the A5 simplex group, B5 cubic group, and the D5 demihypercubic group;
- 58 fundamental Coxeter groups that generate uniform polytopes in the seventh dimension, with only four of these generating uniform non-prismatic figures.
There exist 58 total paracompact Coxeter groups of ranks four through ten, with realizations in dimensions three through nine. These solutions all contain infinite facets and vertex figures, in contrast from compact hyperbolic groups that contain finite elements; there are no other such groups with higher or lower ranks.
- The atomic number of cerium (Ce).
58 is the number of usable cells on a Hexxagon game board.
- 58 is also the partial sum of the first eight records set by highly totient numbers m with values φ(m) = n: {2, 3, 4, 5, 6, 10, 11, 17}.[9]
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