While Masem's suggestion should work, you could do it more simply with loop. For example:

for my $number (1 .. $END) {
  my $logFile = 'pair' . $number . $RUN_NAME . '.log';
  rename $logFile, "$OUT_DIR/allfiles/$logFile";
}
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(You could skip the "my $number" for idiomatic purposes and use $_ instead.)

--PotPieMan