From Gus Wiseman, May 17 2019: 

The a(1) = 1 through a(6) = 17 rulers:

  {0,1}  {0,1,2}  {0,1,3}    {0,1,2,4}    {0,1,2,5}      {0,1,4,6}
                  {0,2,3}    {0,1,3,4}    {0,1,3,5}      {0,2,5,6}
                  {0,1,2,3}  {0,2,3,4}    {0,2,4,5}      {0,1,2,3,6}
                             {0,1,2,3,4}  {0,3,4,5}      {0,1,2,4,6}
                                          {0,1,2,3,5}    {0,1,2,5,6}
                                          {0,1,2,4,5}    {0,1,3,4,6}
                                          {0,1,3,4,5}    {0,1,3,5,6}
                                          {0,2,3,4,5}    {0,1,4,5,6}
                                          {0,1,2,3,4,5}  {0,2,3,5,6}
                                                         {0,2,4,5,6}
                                                         {0,3,4,5,6}
                                                         {0,1,2,3,4,6}
                                                         {0,1,2,3,5,6}
                                                         {0,1,2,4,5,6}
                                                         {0,1,3,4,5,6}
                                                         {0,2,3,4,5,6}
                                                         {0,1,2,3,4,5,6}

The a(1) = 1 through a(6) = 17 compositions are:

  (1)  (11)  (12)   (112)   (113)    (132)
             (21)   (121)   (122)    (231)
             (111)  (211)   (221)    (1113)
                    (1111)  (311)    (1122)
                            (1112)   (1131)
                            (1121)   (1212)
                            (1211)   (1221)
                            (2111)   (1311)
                            (11111)  (2121)
                                     (2211)
                                     (3111)
                                     (11112)
                                     (11121)
                                     (11211)
                                     (12111)
                                     (21111)
                                     (111111)