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


      In mathematics


      58 is a composite number, meaning its factor is 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 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.


      {\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. 58 has an aliquot sum of 32 within an aliquot sequence of two composite numbers (58, 32, 13, 1, 0) in the 13-aliquot tree. There is no solution to the equation



      x
      −
      φ
      (
      x
      )
      =
      58


      {\displaystyle x-\varphi (x)=58}

      , making fifty-eight the sixth noncototient; 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 (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 composite numbers, 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 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).
      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 simple continued fraction 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


      {\displaystyle 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

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