1.333, 1.5, 1.667, 1.75, 2, 3

I didn't find any sequence in a lookup, even for variations.
One remote database find was this "off the wall guess."

If n^(n + 2) + (n + 1)^(n + 3) is Prime, return (n+15)/12

Let's test it (using another program)

Table[If[PrimeQ[n^(n+2)+(n+1)^(n+3)],(n+15)/12.,],{n,21}]

{1.3333333333333333,,1.5,,1.6666666666666665,1.75, ,,2.,,,,,,,,,,,,3.}

Yeah. I know. it's a wild guess!!
The next two solutions would be 3.25, and 4.1666666666

Given the complexity of the prime test.. I'm impressed. Unfortunately, I
don't think this method will qualify as the "formula" Eric was looking for.

Rick