The following set arose in Jonathan Merzel's dissertation (Ph.D. Berkeley, 1980) in an extremely abstract setting. Define the set O(n) recursively, by

O(1) = {1};       O(n) = 2O(n-1) È( O(n-1) + 1)    for n ³ 2.

Here is a short table of O(n), n £ 6:

O(1) = {1} O(2) = {2} O(3) = {3,4} O(4) = {4,5,6, 8} O(5) = {5,6,7,8,9,10, 12,16} O(6) = {6,7,8,9,10,11,12,13,14,16,17,18, 20, 24, 32}

We intend to answer nearly every possible question about O(n).