You bought a big box of random computer parts at a garage sale. You've started putting the parts together to build custom computers.

You want to test the performance of different combinations of parts, and decide to create your own benchmarking program to see how your computers compare. You choose the famous "Sieve of Eratosthenes" algorithm, an ancient algorithm, but one that should push your computers to the limits.

Your task is to create a program that implements the Sieve of Eratosthenes algorithm to find all prime numbers less than or equal to a given number.

A prime number is a number larger than 1 that is only divisible by 1 and itself.
For example, 2, 3, 5, 7, 11, and 13 are prime numbers.
By contrast, 6 is *not* a prime number as it not only divisible by 1 and itself, but also by 2 and 3.

To use the Sieve of Eratosthenes, you first create a list of all the numbers between 2 and your given number. Then you repeat the following steps:

- Find the next unmarked number in your list (skipping over marked numbers). This is a prime number.
- Mark all the multiples of that prime number as
**not**prime.

You keep repeating these steps until you've gone through every number in your list. At the end, all the unmarked numbers are prime.

Note

The tests don't check that you've implemented the algorithm, only that you've come up with the correct list of primes. To check you are implementing the Sieve correctly, a good first test is to check that you do not use division or remainder operations.

Let's say you're finding the primes less than or equal to 10.

- List out 2, 3, 4, 5, 6, 7, 8, 9, 10, leaving them all unmarked.
- 2 is unmarked and is therefore a prime. Mark 4, 6, 8 and 10 as "not prime".
- 3 is unmarked and is therefore a prime.
Mark 6 and 9 as not prime
*(marking 6 is optional - as it's already been marked)*. - 4 is marked as "not prime", so we skip over it.
- 5 is unmarked and is therefore a prime.
Mark 10 as not prime
*(optional - as it's already been marked)*. - 6 is marked as "not prime", so we skip over it.
- 7 is unmarked and is therefore a prime.
- 8 is marked as "not prime", so we skip over it.
- 9 is marked as "not prime", so we skip over it.
- 10 is marked as "not prime", so we stop as there are no more numbers to check.

You've examined all numbers and found 2, 3, 5, and 7 are still unmarked, which means they're the primes less than or equal to 10.

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Deep Dive into Sieve!

We explore various approaches to the Sieve of Eratosthenes, starting with nested loops and lazy evaluation, then moving onto sets and finally looking at recursion.