58 (number)

← 57 58 59 →
← 50 51 52 53 54 55 56 57 58 59 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	fifty-eight
Ordinal	58th (fifty-eighth)
Factorization	2 × 29
Divisors	1, 2, 29, 58
Greek numeral	ΝΗ´
Roman numeral	LVIII
Binary	1110102
Ternary	20113
Senary	1346
Octal	728
Duodecimal	4A12
Hexadecimal	3A16

58 (fifty-eight) is the natural number following 57 and preceding 59.

Mathematics

Form

Fifty-eight is the seventeenth semiprime ^[1] and the ninth with 2 as the lowest non-unitary divisor; thus of the form ${\displaystyle 2\times q}$, where ${\displaystyle q}$ is a higher prime (29).

Number-theoretical

58 is equal to the sum of the first seven consecutive prime numbers: ^[2]

${\displaystyle 2+3+5+7+11+13+17=58.}$

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 ${\displaystyle x-\varphi (x)=58}$, making fifty-eight the sixth noncototient; ^[7] however, the totient summatory function over the first thirteen integers is 58. ^[8]

58 is also the second non-trivial 11-gonal number, after 30. ^[9]

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). ^[10]

More specifically, 58 is the eleventh member in the sequence of consecutive discrete semiprimes that begins, ^[11]

• (14, 15), ^{[lower-alpha 1]}
• (21, 22), ^{[lower-alpha 2]}
• (33, 34, 35), ^{[lower-alpha 3]}
• (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, ^[12] and 22 itself the thirteenth composite. ^[12] (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.

${\displaystyle 58^{17}+1}$ is also semiprime (the second such number ${\displaystyle n}$ for ${\displaystyle n^{17}+1,}$ after 2). ^[13]

Decimal properties

The fifth repdigit is the product between the thirteenth and fifty-eighth primes,

${\displaystyle 41\times 271=11111.}$

58 is also the smallest integer in decimal whose square root has a continued fraction with period 7. ^[14] It is the fourth Smith number whose sum of its digits is equal to the sum of the digits in its prime factorization (13). ^[15]

Mertens function

Given 58, the Mertens function returns ${\displaystyle 0}$, the fourth such number to do so. ^[16] 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. ^[12]

Geometric properties

The regular icosahedron produces fifty-eight distinct stellations, the most of any other Platonic solid, which collectively produce sixty-two stellations. ^{[17] [18]}

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.

Other fields

Base Hexxagōn starting grid, with fifty-eight "usable" cells

58 is the number of usable cells on a Hexxagon game board.

Notes

1. 14 = 2 · 7 and 15 = 3 · 5, where the first four primes are 2, 3, 5, 7.
2. 21 = 3 · 7, and 22 = 2 · 11; factors spanning primes between 2 and 11, aside from 5.
3. 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

1. Sloane, N. J. A. (ed.). "SequenceA001358". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
2. Sloane, N. J. A. (ed.). "SequenceA007504(Sum of the first n primes.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-12-20.
3. Sloane, N. J. A. (ed.). "SequenceA000040(The prime numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-12-20.
4. Sloane, N. J. A. (ed.). "SequenceA006450(Prime-indexed primes: primes with prime subscripts.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-12-20.
5. Sloane, N. J. A. (ed.). "SequenceA001065(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.
6. Sloane, N. J. A., ed. (1975). "Aliquot sequences". Mathematics of Computation. 29 (129). OEIS Foundation: 101–107. Retrieved 2024-02-27.
7. "Sloane's A005278 : Noncototients". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
8. Sloane, N. J. A. (ed.). "SequenceA002088(Sum of totient function.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-02-27.
9. "Sloane's A051682 : 11-gonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
10. Sloane, N. J. A. (ed.). "SequenceA001358(Semiprimes (or biprimes): products of two primes.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-02-27.
11. Sloane, N. J. A. (ed.). "SequenceA006881(Semiprimes (or biprimes): products of two primes.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-05-07.
12. Sloane, N. J. A. (ed.). "SequenceA002808(The composite numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-05-07.
13. Sloane, N. J. A. (ed.). "SequenceA104494(Positive integers n such that n^17 + 1 is semiprime.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-02-27.
14. "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.
15. "Sloane's A006753 : Smith numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
16. "Sloane's A028442 : Numbers n such that Mertens' function is zero". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
17. 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.
18. Webb, Robert. "Enumeration of Stellations". Stella. Archived from the original on 2022-11-26. Retrieved 2023-01-18.