Hello Math-Fun (we need help to check, compute and submit this to the OEIS in good mathematical English — as we fear Dan's furrowed brow ;-D) This sequence is based on the Aware strategic game. We empty a “pit” and sow to the right its content – one seed per pit. When this is done, we continue from there, empty the next pit and sow again one seed per pit to the right. S0 = 1, 109, 2, 8, 27, 3, 26, 31, 5, 4, 30, 53, 29, 52, 6, 28, 51, 61, 7, 50, 60, 9, 749, 59, 96, 48, 58, 95, 47, 57, 94, 10, … To start, we empty the first pit (1) and sow its unique seed to the right; 109 becomes 110 and S0 becomes S1 (the vertical stroke shows where we are in the process): S1 = _, 110, | 2, 8, 27, 3, 26, 31, 5, 4, 30, 53, 29, 52, 6, 28, 51, 61, 7, 50, 60, 9, 749, 59, 96, 48, 58, 95, 47, 57, 94, 10, … We now empty the first “untouched” pit (2) and sow its content in the same way. S1 becomes: S2 = _, 110, _, 9, 28, | 3, 26, 31, 5, 4, 30, 53, 29, 52, 6, 28, 51, 61, 7, 50, 60, 9, 749, 59, 96, 48, 58, 95, 47, 57, 94, 10, … We proceed like this, step by step, ad infinitum: S3 = _, 110, _, 9, 28, _, 27, 32, 6, | 4, 30, 53, 29, 52, 6, 28, 51, 61, 7, 50, 60, 9, 749, 59, 96, 48, 58, 95, 47, 57, 94, 10, … S4 = _, 110, _, 9, 28, _, 27, 32, 6, _, 31, 54, 30, 53, | 6, 28, 51, 61, 7, 50, 60, 9, 749, 59, 96, 48, 58, 95, 47, 57, 94, 10, … S5 = _, 110, _, 9, 28, _, 27, 32, 6, _, 31, 54, 30, 53, _, 29, 52, 62, 8, 51, 61, | 9, 749, 59, 96, 48, 58, 95, 47, 57, 94, 10, … S6 = _, 110, _, 9, 28, _, 27, 32, 6, _, 31, 54, 30, 53, _, 29, 52, 62, 8, 51, 61, _, 750, 60, 97, 49, 59, 96, 48, 58, 95, | 10, … If we now compare the end result Se to the starting sequence S0, we see that… S0 = 1, 109, 2, 8, 27, 3, 26, 31, 5, 4, 30, 53, 29, 52, 6, 28, 51, 61, 7, 50, 60, 9, 749, 59, 96, 48, 58, 95, 47, 57, 94, 10 … Se = _, 110, _, 9, 28, _, 27, 32, 6, _, 31, 54, 30, 53, _, 29, 52, 62, 8, 51, 61, _, 750, 60, 97, 49, 59, 96, 48, 58, 95, _, … … S0 and Se share the same digit succession. S0 should be the lexicographically earliest sequence of distinct positive integers with this property. This is not too tricky, I guess – still, I hope I didn't leave too many mistake above (and that this is not old hat). Best, É.