58 (fifty-eight) is the

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

following 57 and preceding 59.

In mathematics

58 is a

composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime numb ...

with four factors: 1, 2, 29, and 58. Other than 1 and the number itself, 58 can be formed by multiplying two primes 2 and 29, making it a semiprime. 58 is not divisible by any

square number In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...

other than 1, making it a square-free integer A semiprime that is not square numbers is called a squarefree semiprime, and 58 is among them. 58 is equal to the sum of the first seven consecutive prime numbers: :

2 + 3 + 5 + 7 + 11 + 13 + 17 = 58.

This is a difference of 1 from the seventeenth prime number and seventh super-prime, 59. 58 has an

aliquot sum In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of , that is, all divisors of other than itself. That is, s(n)=\sum_ d \, . It can be used to characterize the prime numbers, perfect numbers, sociabl ...

of 32 within an

aliquot sequence In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0. Def ...

of two composite numbers (58, 32, 31, 1, 0) in the 31-aliquot tree. There is no solution to the equation

x - \varphi(x) = 58

, making fifty-eight the sixth

noncototient In number theory, a noncototient is a positive integer that cannot be expressed as the difference between a positive integer and the number of coprime integers below it. That is, , where stands for Euler's totient function In number theory ...

; however, the totient summatory function over the first thirteen integers is 58. On the other hand, the Euler totient of 58 is the second

perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfec ...

( 28), 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. 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

s, and 22 itself the thirteenth composite. (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 factorization In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a comp ...

s that nearly span a set of consecutive prime numbers.

58^ + 1

is also semiprime (the second such number

n

for

n^ + 1,

after 2). The fifth repdigit is the product between the thirteenth and fifty-eighth primes, :

41 \times 271 = 11111.

58 is also the smallest integer in

decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of th ...

whose

square root In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...

has a

simple continued fraction A simple or regular continued fraction is a continued fraction with numerators all equal one, and denominators built from a sequence \ of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated) continued fr ...

with period 7. It is the fourth Smith number whose sum of its digits is equal to the sum of the digits in its prime factorization (13). Given 58, the Mertens function returns

0

, the fourth such number to do so. The sum of the first three numbers to return zero (2, 39, 40) sum to 81 = 9², which is the fifty-eighth composite number.

Notes

References

{{Integers, zero Integers