For the first unused "$name$n.txt" file: 
if (-f "$name.txt") {
    $n=1;
    $n++ while -f "$name$n.txt";
}

[download]
For one greater than the highest "$name$n.txt" file, something like kcott's solution below works.

In reply to Re: Sequentially numbering output files rather than overwriting an existing file by educated_foo
in thread Sequentially numbering output files rather than overwriting an existing file by TJCooper

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