Recursive sum of each decimal places. This output is in 9 cycle.

recursive-sum-digit.pl

#!/usr/bin/perl

use strict;
use warnings;
use constant MAX_NUMBER => 1_000_000;
use List::Util qw(sum);

my %stat;

for (my $i = 1; $i < MAX_NUMBER; $i++) {
    my $recursive_sum_digit = recursive_sum_digit($i);
    print "$i => $recursive_sum_digit\n";
    $stat{$recursive_sum_digit}++;
}

# stat
printf qq{%d:%d(%2.2f%%) }, $_, $stat{$_}, $stat{$_}/MAX_NUMBER for 1..9;
print "\b\n";

sub recursive_sum_digit {
    my $number = sum_digit(shift);
    while ($number >= 10)  {
        $number = sum_digit($number);
    }
    return $number;
}

sub sum_digit {
    my $number = shift;
    #print "number=$number\n";
    return sum split //, $number;
}