Problem 52567. Easy Sequences 7: Easy as Composite Pi

Created by Ramon Villamangca

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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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Last Solution submitted on Nov 23, 2022

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3 Comments

Lincoln Poon on 25 Oct 2022

I mean...doing this above 1e10 won't be easy anyway, no matter prime or composite.

GeeTwo on 26 Oct 2022

Lincoln Poon,
Yes, they're really the same problem, as (designating the composite counting function as κ(n)), κ(n) + 𝜋(n) + 1 = n. There are techniques to do the count without creating an array for x>√n, and even for x>∛n.

Rafael S.T. Vieira on 15 Nov 2022

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.

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Problem 52567. Easy Sequences 7: Easy as Composite Pi

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Problem 52567. Easy Sequences 7: Easy as Composite Pi

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