05AB1E, 25 bytes
æ¦ε.¿DIāÉ-s÷D>*2÷Æ®ygmP}O æ¦ε } # map over each non-empty subset of the divisors .¿D # take the GCDLCM of the subset IāÉ- # push the range with the lower bound decreased s÷ # integer divide modified range by GCDLCM D>*2÷ # for both resulting integers, compute the nth triangular number Æ # reduce by subtraction (results in negative values) ®ygm* # multiply with GCDLCM and (-1)**len(subset) O # sum the resulting list