## Number of primes less than given integer

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• 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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