Not really an answer but too long for a comment.

The numbers start **1,23, 6, 6, 12, 15, 30, 30, 60, 60, 84, 105, 210, 210, 420, 420, 420, 420, 840, 840, 1260**, 1260, 2310, 2310, 4620, 4620, 5460, 5460, 9240, 9240, 13860, 13860, 16380, 16380, 30030, 30030, 60060, 60060

and the bold terms match (up to some leading 0's)  [largest LCM of a 5-partition of n][1]. This sequence is not there exactly as is. However it is pretty close to  [largest lcm of partitions of n][2] with a few hiccups. That is because the best partitions are the same most of the time and cosist of the first few primes with perhaps 4,8 or 9 included. Sometimes a prime near the end is skipped to get to the next prime. I imagine that after a partition like $[2,3,5,7,\cdots,89]$ things might get a bit more exciting since $91,93,95$ are all composite. 


  [1]: http://oeis.org/A129650
  [2]: http://oeis.org/A000793