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, 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.

In mathematics

58 is a composite number with four factors: 1, 2, 29, and 58. ^[1] Other than 1 and the number itself, 58 can be formed by multiplying two primes 2 and 29, making it a semiprime. ^[2] 58 is not divisible by any square number other than 1, making it a square-free integer ^[3] A semiprime that is not square numbers is called a squarefree semiprime, and 58 is among them. ^[4]

58 is equal to the sum of the first seven consecutive prime numbers: ^[5]

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

This is a difference of 1 from the seventeenth prime number and seventh super-prime, 59. ^{[6] [7]} 58 has an aliquot sum of 32 ^[8] within an aliquot sequence of two composite numbers (58, 32, 31, 1, 0) in the 31-aliquot tree. ^[9] There is no solution to the equation ${\displaystyle x-\varphi (x)=58}$, making fifty-eight the sixth noncototient; ^[10] however, the totient summatory function over the first thirteen integers is 58. ^{[11] [a]}

On the other hand, the Euler totient of 58 is the second perfect number (28), ^[13] 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. ^[14]

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, ^[15] and 22 itself the thirteenth composite. ^[15] (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). ^[16]

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 simple continued fraction with period 7. ^[17] It is the fourth Smith number whose sum of its digits is equal to the sum of the digits in its prime factorization (13). ^[18]

Given 58, the Mertens function returns ${\displaystyle 0}$, the fourth such number to do so. ^[19] 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. ^[15]

Notes

1. 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}. ^[12]

References

1. Anjema, Henry (1767). Table of divisors of all the natural numbers from 1. to 10000. p.  3. ISBN   9781140919421 – via the Internet Archive.
2. Neil, Sloane; Guy, R. K. (22 August 2010). "A001358: Semiprimes (or biprimes): products of two primes". On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 April 2024.
3. Sloane, Neil (n.d.). "A005117: Squarefree numbers: numbers that are not divisible by a square greater than 1". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 April 2024.
4. 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.
5. 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.
6. Sloane, N. J. A. (ed.). "SequenceA000040(The prime numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-12-20.
7. 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.
8. 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.
9. Sloane, N. J. A., ed. (1975). "Aliquot sequences". Mathematics of Computation. 29 (129). OEIS Foundation: 101–107. Retrieved 2024-02-27.
10. "Sloane's A005278 : Noncototients". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
11. Sloane, N. J. A. (ed.). "SequenceA002088(Sum of totient function.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-02-27.
12. Sloane, N. J. A. (ed.). "SequenceA131934(Records in A014197.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-07-02.
13. Sloane, N. J. A. (ed.). "SequenceA000010(Euler totient function.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-07-02.
14. "Sloane's A051682 : 11-gonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
15. Sloane, N. J. A. (ed.). "SequenceA002808(The composite numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-05-07.
16. 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.
17. "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.
18. "Sloane's A006753 : Smith numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
19. "Sloane's A028442 : Numbers n such that Mertens' function is zero". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.