Lengths of the lists L(m):

1, 1, 1, 2, 2, 2, 4, 3, 2, 3, 4, 5, 5, 3, 5, 7, 5, 4, 6, 7, 7, 7, 5, 8, 9, 5, 6, 9, 7, 7, 10, 7, 7, 9, 9, 7, 8, 8, 11, 10, 6, 8, 13, 9, 6, 9, 12, 10, 8, 7, 12, 12, 7, 8, 13, 11, 10, 12, 11, 11, 12, 11, 11, 12, 11, 13, 14, 9, 11, 15, 13, 10, 12, 14, 14, 12, 8, 14, 16, 10, 9, 15, 15, 14, 14, 11, 15, 16, 12, 10, 16, 15, 14, 13, 12, 15, 16, 11, 11, 16, 15