It is conjectured that this list is complete.
Comments from Richard C. Schroeppel:
"I can't put my hand on the book, but I think Vaughn has shown
that every sufficiently large number is of the form a^2+b^3+c^5.
The key is that the sum 1/2 + 1/3 + 1/5 = 31/30 > 1, so the expected
number of representations for a number N is, on average, K * N^1/30.
K is some mess of gamma(fractions). [It's also necessary to check
that there's no modular exclusion.] I once tried to estimate how
large an N was "sufficiently large" and if I remember correctly got around 10^60.
The results for a^2+b^3+c^4+d^5 and a^2+b^3+c^4+d^5+e^6 would
follow immediately, although proving an upper bound is a big
question. I'm not even sure if Vaughn's proof is constructive.
Christopher Landauer worked on the 2...6 problem long long ago;
I think his program got up to a million or so.
There was a paper in Math. Comp. about five years ago, about the
problem of 4 cubes. Beeler & I tried to find the empirical last-
unrepresentable number, but couldn't reach it. The Math. Comp. paper found
a likely candidate. It was around 10^13 or 14."
No other n < 10^8. - T. D. Noe, Mar 10 2008