58 (number)

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

58 is a composite number, meaning its factor is 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, 13, 1, 0) in the 13-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 = 9², which is the fifty-eighth composite number.^[15]

• The atomic number of cerium (Ce).

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