Ternary analogue of A094913?

[Jonathan Post]

Excellent work, Maximilian, Giovanni, Neil!

I've actually used this line in poetry, and in my oration at my
father's funeral:

"Every word begins and ends with the empty word.
The empty word begins and ends with itself."

Giovanni's lexicographically first examples begin with substrings of
zeros, which would be removed by OEIS default notation, so what are
the first examples of each maximum which do not begin with 0?

What  Maximilian found, after correctly adding the empty words, that
relates this function to the ratio of symmetric functions on primes is
very nice.  I'm not clear on A077168, but if so, then my innocent
question has led OEIS experts to a do the heavy lifting of finding a
deeper connection between several sequences.

That none of this was obvious beforehand, although plausible when
pointed out to me, suggests a 1-page paper in, say, Mathematics
Magazine, for students and teachers, with Maximilian as primary
author.

Best,

Jonathan Vos Post