Formulae for complementary sequences

[David Wilson]

On A000037 (the nonsquares), there is the formula

a(n) = n + [sqrt(n + [sqrt(n)])]

On A007412 (the noncubes), we find

a(n) = n + [cbrt(n + [cbrt(n)])]

I also tried

a(n) = n + [log2(n + [log2(n)])]

and with the exception of a(1) = 1, got the non-powers-of-2.

When I tried n + [log(n + [log(n)])]

I got the number not of the form ceil(exp(k)), again with initial
exceptions.

I would assume that for sufficiently fast-growing f

a(n) = n + [f^-1(n + [f^-1(n)])

is a formula for the numbers not of the form ceil(f(k)), except
for possibly a few initial terms.  Is this the case?

- David W. Wilson

"Truth is just truth -- You can't have opinions about the truth."
   - Peter Schickele, from P.D.Q. Bach's oratorio "The Seasonings"