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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
58 (fifty-eight ) is the natural number following 57 and preceding 59 .
Mathematics
[ edit ]
[ edit ]
Fifty-eight is the seventeenth semiprime [1] and the ninth with 2 as the lowest non-unitary divisor ; thus of the form
2
×
q
{\displaystyle 2\times q}
, where
q
{\displaystyle q}
is a higher prime (29 ).
Number-theoretical
[ edit ]
58 is equal to the sum of the first seven consecutive prime numbers:[2]
2
+
3
+
5
+
7
+
11
+
13
+
17
=
58.
{\displaystyle 2+3+5+7+11+13+17=58.}
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
x
−
φ
(
x
)
=
58
{\displaystyle x-\varphi (x )=58}
, making fifty-eight the sixth noncototient ;[7] however, the totient summatory function over the first thirteen integers is 58.[8] [a]
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
[ edit ]
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 ),[b]
(21, 22),[c]
(33, 34, 35),[d]
(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.
58
17
+
1
{\displaystyle 58^{17}+1}
is also semiprime (the second such number
n
{\displaystyle n}
for
n
17
+
1
,
{\displaystyle n^{17}+1,}
after 2 ).[15]
Decimal properties
[ edit ]
The fifth repdigit is the product between the thirteenth and fifty-eighth primes,
41
×
271
=
11111.
{\displaystyle 41\times 271=11111.}
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
[ edit ]
Given 58, the Mertens function returns
0
{\displaystyle 0}
, the fourth such number to do so.[18] The sum of the first three numbers to return zero (2, 39 , 40 ) sum to 81 = 9 2 , which is the fifty-eighth composite number.[14]
Geometric properties
[ edit ]
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
[ edit ]
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 A 5 simplex group, B 5 cubic group, and the D 5 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.
In science
[ edit ]
Other fields
[ edit ]
Base Hexxagōn starting grid, with fifty-eight "usable" cells
58 is the number of usable cells on a Hexxagon game board.
Notes
[ edit ]
^ 14 = 2 · 7 and 15 = 3 · 5, where the first four primes are 2, 3, 5, 7.
^ 21 = 3 · 7, and 22 = 2 · 11; factors spanning primes between 2 and 11 , aside from 5 .
^ 33 = 3 · 11, 34 = 2 · 17, and 35 = 5 · 7; in similar form, a set of factors that are the primes between 2 and 17, aside from 13 ; the last such set of set of prime factors that nearly covers consecutive primes.
References
[ edit ]
^ Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-12-20 .
^ Sloane, N. J. A. (ed.). "Sequence A006450 (Prime-indexed primes: primes with prime subscripts.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-12-20 .
^ Sloane, N. J. A. (ed.). "Sequence A001065 (Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-02-27 .
^ Sloane, N. J. A. , ed. (1975). "Aliquot sequences" . Mathematics of Computation . 29 (129). OEIS Foundation: 101–107. Retrieved 2024-02-27 .
^ "Sloane's A005278 : Noncototients" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-30 .
^ Sloane, N. J. A. (ed.). "Sequence A002088 (Sum of totient function.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-02-27 .
^ Sloane, N. J. A. (ed.). "Sequence A131934 (Records in A014197.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-07-02 .
^ Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-07-02 .
^ "Sloane's A051682 : 11-gonal numbers" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-30 .
^ Sloane, N. J. A. (ed.). "Sequence A001358 (Semiprimes (or biprimes): products of two primes.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-02-27 .
^ Sloane, N. J. A. (ed.). "Sequence A006881 (Semiprimes (or biprimes): products of two primes.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-05-07 .
^ a b c Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-05-07 .
^ Sloane, N. J. A. (ed.). "Sequence A104494 (Positive integers n such that n^17 + 1 is semiprime.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-02-27 .
^ "Sloane's A013646: Least m such that continued fraction for sqrt(m ) has period n " . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2021-03-18 .
^ "Sloane's A006753 : Smith numbers" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-30 .
^ "Sloane's A028442 : Numbers n such that Mertens' function is zero" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-30 .
^ H. S. M. Coxeter ; P. Du Val; H. T. Flather; J. F. Petrie (1982). The Fifty-Nine Icosahedra . New York: Springer. doi :10.1007/978-1-4613-8216-4 . ISBN 978-1-4613-8216-4 .
^ Webb, Robert. "Enumeration of Stellations" . Stella . Archived from the original on 2022-11-26. Retrieved 2023-01-18 .
t
e
100,000
1,000,000
10,000,000
100,000,000
1,000,000,000
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