Prime(n) and pi(n) are inverse functions: A000720(a(n)) = n and a(n) is the least number m such that a( A000720(m)) = a(n). This shows that there exist infinitely many prime numbers.' - Pieter Moree, Oct 14 2004ġ is not a prime, for if the primes included 1, then the factorization of a natural number n into a product of primes would not be unique, since n = n*1. The paper by Kaoru Motose starts as follows: 'Let q be a prime divisor of a Mersenne number 2^p-1 where p is prime. For contributions concerning 'almost primes' see A002808.Ī number p is prime if (and only if) it is greater than 1 and has no positive divisors except 1 and p.Ī natural number is prime if and only if it has exactly two (positive) divisors.Ī prime has exactly one proper positive divisor, 1. For all information concerning prime powers, see A000961. Once we calculate all the multiples through 7 we can be sure that we've generated all of the multiples. Then in our inner loop we will produce range(6, 50, 3) which will add 6, 9, 12, 15, etc to our noprimes set.Īs you can see, this simply generates multiples.

The next time through we generate the number 3. Each of these numbers will be added to our noprimes set. This produces: 4, 6, 8, 10, etc until we hit 50.

Then we iterate through the inner loop using 2 to produce this range: range(4, 50, 2). Hopefully that isn't too complicated! As an example, the first time we iterate through the outer loop, we generate the number 2. During each iteration we will use the number to iterate through the range 2i through 50 with an interval of i. In this example we are using set comprehension to iterate through the numbers 2 through 7. The function range(2, 8) will generate the numbers 2-7. We are using a set in this case because we only want to include each multiple once. noprimes = set(j for i in range(2, 8) for j in range(i*2, 50, i)) Therefore, if we want to generate a list of non-primes under 50 we can do so by generating multiples. A prime number is one that is only divisible by 1 and itself. To begin, we need to think about what a prime number is.

Generating a list of non-primes is much simpler than generating a list of primes.

List of prime numbers less than 100