The prime Pi function is defined as the number of prime numbers from 1 to a certain given limit. In MATLAB, it is easy to create a prime Pi procedure, because there are built-in functions such as "primes" and "isprime". To calculate the prime Pi up to 100, we may just proceed as follows:
>> numel(primes(100))
>> ans =
25
>> nnz(isprime(1:100))
>> ans =
25
Can we make a function for "composite Pi", which is the number of composite numbers from 1 to a given limit, inclusive? Let's find out...
NOTE: The number '1' is considered as neither prime nor composite.
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I mean...doing this above 1e10 won't be easy anyway, no matter prime or composite.
Lincoln Poon,
Yes, they're really the same problem, as (designating the composite counting function as κ(n)), κ(n) + ?(n) + 1 = n. There are techniques to do the count without creating an array for x>√n, and even for x>∛n.
If anyone does find a way to solve this precisely without some hack, please, publish a scientific paper, and do not post your code here. This problem is not made from an easy sequence at all.