[Euler] Add solution to No. 23.

Slow on rakudo-jvm but it works.

Author: shlomif

euler/prob023-shlomif.pl

@@ -0,0 +1,78 @@
+#!/usr/bin/perl
+# By Shlomi Fish.
+#
+# Based on:
+# https://bitbucket.org/shlomif/project-euler/src/aa5eecd18f0825901afeb3c54dcda0da79ac3576/project-euler/23/euler-23-4.pl?at=default
+#
+# Solution for:
+#
+# http://projecteuler.net/problem=23
+#
+# A perfect number is a number for which the sum of its proper divisors
+# is exactly equal to the number. For example, the sum of the proper
+# divisors of 28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 28
+# is a perfect number.
+#
+# A number n is called deficient if the sum of its proper divisors is
+# less than n and it is called abundant if this sum exceeds n.
+#
+# As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the
+# smallest number that can be written as the sum of two abundant numbers
+# is 24. By mathematical analysis, it can be shown that all integers
+# greater than 28123 can be written as the sum of two abundant numbers.
+# However, this upper limit cannot be reduced any further by analysis
+# even though it is known that the greatest number that cannot be
+# expressed as the sum of two abundant numbers is less than this limit.
+#
+# Find the sum of all the positive integers which cannot be written as
+# the sum of two abundant numbers.
+#
+
+my @divisors_sums;
+@divisors_sums[1] = 0;
+
+my $MAX = 28_123;
+for (1 .. ($MAX +> 1)) -> $div
+{
+    loop (my $prod = ($div +< 1); $prod <= $MAX; $prod += $div)
+    {
+        @divisors_sums[$prod] += $div;
+    }
+}
+
+# Memoized.
+#
+my @is_abundant_sum;
+
+my @abundants;
+my $total = 0;
+for (1 .. $MAX) -> $num
+{
+    if @divisors_sums[$num] > $num
+    {
+        @abundants.push($num);
+        # The sub { ... } and return are a workaround for the fact that Rakudo
+        # Perl 6 does not have last LABEL yet.
+        my $c = sub {
+            for @abundants -> $i
+            {
+                if ((my $s = $i + $num) > $MAX)
+                {
+                    return;
+                }
+                else
+                {
+                    if (! @is_abundant_sum[$s])
+                    {
+                        $total += $s;
+                        @is_abundant_sum[$s] = True;
+                    }
+                }
+            }
+        };
+
+        $c();
+    }
+}
+
+say "Sum == ", ((((1 + $MAX) * $MAX) +> 1)-$total);