PWC150 - Square-free Integer


On with TASK #2 from The Weekly Challenge #150. Enjoy!

The challenge

Write a script to generate all square-free integers <= 500.

In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no perfect square other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, 10 = 2 ⋅ 5 is square-free, but 18 = 2 ⋅ 3 ⋅ 3 is not, because 18 is divisible by 9 = 3**2.


The smallest positive square-free integers are
    1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, ...

The questions

As it often happens, I’m nitpicking on the details about the domain of our investigation: should we consider negative values? I guess not by the examples…

The solution

We will start with Perl first:

sub is_square_free ($N) {
   return unless $N % 4;
   my $divisor = 3;
   while ($N > $divisor) {
      if ($N % $divisor == 0) {
         $N /= $divisor;
         return unless $N % $divisor;
      $divisor += 2; # go through odd candidates only
   return 1;

The goal is not to find all divisors, so… we don’t find them and we take every possible chance to bail out with a false value. It can happen in two cases:

  • if the number is a multiple of 4 because… 4 is a square, you know;
  • otherwise, if the number happens to have the same divisor twice.

Why the explicit check on 4? Well, in this way we can get the prime number 2 out of the way, and iterate only through odd divisors, starting at 3. Actually, we might start at 7 because the first positive integer that is neither a multiple of 4 nor square-free is 9. Whatever.

I like the Raku translation better because it allows us to use the is-divisible-by operator %%, instead of its “contrary” (sort of) remainder-in-the-division-by %:

sub is-square-free ($N is copy) {
   return False if $N %% 4;
   my $divisor = 3;
   while $N > $divisor {
      if $N %% $divisor {
         $N = ($N / $divisor).Int;
         return False if $N %% $divisor;
      $divisor += 2; # go through odd candidates only
   return True;

This makes the whole thing more readable, but at the end of the day it was pretty readable also to begin with. I have a little itch in the fact that the division between the two integers gives out a rational even when the result is an integer… but whatever.

I also like the availability of proper boolean constants, again I think it adds to the readability.

The Raku version also allowed me to play a bit with multi subroutines, in the MAIN:

multi sub MAIN (Int $limit = 500) {
   my @list = (1 .. $limit).grep({is-square-free($_)});
   while @list {
      @list.splice(0, 20).join(', ').print;
      put @list ?? ',' !! '';

multi sub MAIN (*@args) {
   put $_, ' ', (is-square-free($_) ?? 'is' !! 'is not'), ' square free'
      for @args;

I’m providing three different ways to call the program:

  • with no parameter, the limit is set to 500 like the challenge asks;
  • with one single parameter, the limit is set by the parameter itself;
  • with multiple parameters, each is checked for being square-free or not.

The multi helps distinguishing the first two cases from the last, which is functionally nifty.

OK, I didn’t include the full programs for both languages… but you know where to find them should you be curious.

Stay safe and have fun!

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