factorizations n gives the prime factorizations of numbers in [1..n]. Each factorization is sorted.

factorizations 50 == Map'.fromList [(n, primeFactors n) | n <- [1..50]]

>>> take 9 factorizationsAsc
[[],[2],[3],[2,2],[5],[2,3],[7],[2,2,2],[3,3]]

take 50 factorizationsAsc == (primeFactors <$> [1..50])

This global constant is an infinite list of prime numbers. It is generated by a lazy wheel sieve and shared across the whole program run. If you are concerned about the memory requirements of sharing many primes you can call the function wheelSieve directly.

Checks whether a given number is prime.

This function uses trial division to check for divisibility with all primes below the square root of the given number. It is impractical for numbers with a very large smallest prime factor.

forAll (listOf $ elements $ take 20 primes) (\fs -> primeFactors (product fs) == sort fs)

\(Positive n) -> isSorted (primeFactors n)

smallestPrimeFactor n is the smallest prime factor p such that p divides n.

largestPrimeFactor n is the largest prime factor p such that p divides n.

>>> largestPrimeFactor 2
2

>>> largestPrimeFactor 3
3

>>> largestPrimeFactor 4
2

>>> largestPrimeFactor 99
11

forAll (elements $ take 50 primes) (\p -> largestPrimeFactor p == p)

The divisors of a number, in no particular order.

countDivisors n is the number of integers d from [1..n] such that d divides n.

The divisors of product fs, where fs are all prime, in no particular order.

The proper divisors of a number, in no particular order.

Example from Euler problem 21:

>>> sort (properDivisors 220)
[1,2,4,5,10,11,20,22,44,55,110]

The proper divisors of product fs, where fs are all prime, in no particular order.