Formulae for complementary sequences

[Paul D Hanna]

I believe the numbers not of the form ceil(f(k)) 
is given by the recursion: 
 
  a(n) = n + [f^-1(a(n))]
 
Fast growing functions f() require only a few 
nestings for all n > some integer m. 
 
Paul

-- "David Wilson" <davidwwilson at comcast.net> wrote:
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"