

Definition

A factorion is a natural number that equals the sum of the factorials of its digits.



Example
 
145   is a factorion in base 10 because:


          1! + 4! + 5!   =   1 + 24 + 120   =   145 





It can be shown (see talk page) that no factorion in base 10 can exceed   1,499,999.



Task

Write a program in your language to demonstrate, by calculating and printing out the factorions, that:

   There are   3   factorions in base   9
   There are   4   factorions in base 10
   There are   5   factorions in base 11 
   There are   2   factorions in base 12     (up to the same upper bound as for base 10)


See also

 Wikipedia article
 OEIS:A014080 - Factorions in base 10
 OEIS:A193163 - Factorions in base n


constant @factorial = 1, |[\*] 1..*;
 
constant $limit = 1500000;
 
constant $bases = 9 .. 12;
 
my @result;
 
$bases.map: -> $base {
 
    @result[$base] = "\nFactorions in base $base:\n1 2";
 
    sink (1 .. $limit div $base).map: -> $i {
        my $product = $i * $base;
        my $partial;
 
        for $i.polymod($base xx *) {
            $partial += @factorial[$_];
            last if $partial > $product
        }
 
        next if $partial > $product;
 
        my $sum;
 
        for ^$base {
            last if ($sum = $partial + @factorial[$_]) > $product + $_;
            @result[$base] ~= " $sum" and last if $sum == $product + $_
        }
    }
}
 
.say for @result[$bases];