Using a recursive function will always eat up more memory as you go, especially since your functions never stop or return anything. I suppose out your two examples, the second one is better, just because it's not recursive. Not that you asked, but I think you could write it more clearly and elegantly like this:

#!/usr/bin/perl -w
use strict;

while ( <STDIN> ) {
    chomp();

    if ( /^\d+$/ ) {
        print "You entered: $_\n";
    }
    else {
        print "Not a number!\n";
    }
}
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