$artist_link->url() =~ /(\d{9})/;
print $pro_list $1, "\n";
[download]

should help.

Ask if you don't understand some parts of my script.

That doesn't catch cases when the regular expression doesn't match and hence $1 might have a value assigned to it from a previous successful capturing regular expression match.

I humbly suggest either:

print {$pro_list} $1, "\n" if $artist_link->url() =~ /(\d{9})/;
[download]

or

if ($artist_link->url() =~ /(\d{9})/) {
    print {$pro_list} $1, "\n";
}
[download]

The curly braces around {$pro_list} disambiguates its use as the filehandle that is printed to.

We selected all links whose URL matches the regexp, then we loop through tnem. I'd say it always matches.