How many kinds of numbers does n*n multiplication table have? What is its order?

ntable-numbers.pl

#!/usr/bin/perl

use strict;
use warnings;

use Memoize qw(memoize);

memoize("get_variation_hash");

for my $n (9..1_000_000) {
    my $variation       = get_variation_hash($n);
    my $variation_count = scalar keys %$variation;
    #printf "%4d: %4d\n", $n, $variation_count;
    #printf "%4d: %4d (%.4f)\n", $n, $variation_count, $variation_count/log($n);
    #printf "%4d: %4d (%.4f)\n", $n, $variation_count, $variation_count/($n*$n*log($n));
    printf "%4d: %4d (%.4f)\n", $n, $variation_count, $variation_count/($n*$n);
}

sub get_variation_hash {
    my $n = shift;
    if ( $n < 1 || $n =~ /\D/ ) {
        die "n は 1 以上の自然数である必要があります";
    } elsif ( $n == 1 ) {
        return +{ 1 => 1 }; # 1 という数字がある（フラグ1）だけ
    }
    my $variation = \%{ get_variation_hash($n-1) };
    for my $i (1..$n) {
        $variation->{ $i * $n }++;
    }
    return $variation;
}