[seqfan] Sequence needs analysis: A277144: lex earliest avoiding 5/4-th powers

[Neil Sloane]

Dear Seq Fans,  Eric Rowland gave a lovely talk at Rutgers on Thursday and
at my urging has added a bunch of sequences from

Lara Pudwell and Eric Rowland, Avoiding fractional powers over the natural
numbers, arXiv:1510.02807 [math.CO] (2015)

to the OEIS. They were able to find recurrences for many of them.  The
first one
whose structure is not known is A277144.  There is an "a-file" there that
has the first 400,000 terms.

Can anyone find a recurrence or other structure for this sequence?

(For examples of the kind of structure that might exist, check
the other sequences that Eric has uploaded - look for entries that mention
their arXiv paper.  But my hunch is that there might be a totally different
structure to the sequence, one that they did not consider, so don't be
constrained by what they did!)