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Number of primes less than given integer

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  • Kermit Rose
    Hello. The question about prime number distribution and the even = sum of two primes conjecture reminded me of the following. Primes less than 30 are:
    Message 1 of 1 , Aug 29, 2009
      Hello.

      The question about prime number distribution and the
      even = sum of two primes conjecture

      reminded me of the following.


      Primes less than 30 are:

      {2,3,5,7,11,13,17,19,23,29}

      There are 10 of them

      30 + Number of positive primes < square root of 30,
      - one
      - number of even positive integers less than or equal to 30
      - number of multiples of three less than or equal to 30
      + number of multiples of six less than or equal to 30
      - number of multiples of five less than or equal to 30
      + number of multiples of ten less than or equal to 30
      + number of multiples of fifteen less than or equal to 30
      - number of multiples of thirty less than or equal to 30,

      = number of positive primes less than or equal to 30.

      30 + 3 - 1 - 30/2) - 30/3) + 30/6
      - 30/5) + 30/10 + 30/15 - 30/30

      = 30 - 1 + 1 - 15 + 1 - 10 + 5 + 1 - 6 + 3 + 2 -1
      = 10


      Let n be a positive integer.

      Let m be the integer part of the square root of n.

      Define the function

      Mu(J)

      = 1 if J is the product of an even number of distinct prime factors,
      = -1 if J is the product of an odd number of distinct prime factors,
      = 0 if J is divisible by a square > 1.

      Then the number of primes less than or equal to n
      is
      n - 1 + number of primes less than or equal to m
      + sum( for J = 2 to n, mu(J) * int(n/J) ).


      Kermit
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