Asymptotics about A005228

[David W. Wilson]

Let f = A005228, and let g = complement of f.

 

The fact that f and g are complements leads to

 

    [1]  #{k: f(k) <= n} + #{k: g(k) <= n} = n

 

And the fact that f is the running sum of g leads to

 

    [2]  f(n) = 1 + SUM(k < n; g(k))

 

In order to guess at the asymptotics of f, we handwave f and g into real
functions and freely translate [1] and [2] into similar real relationships,
getting:

 

    [3]   f^-1(x) + g^-1(x) = x   where f^-1 is the inverse of f

 

    [4]  f = g'

 

This leads to

 

    [5]  f^-1(x) + (f')^-1(x) = x

 

Of course, I have blithely ignored a lot of details, but one might hope that
a function satisfying [5] would give a good asymptotic to A005228.

 

I wish I could solve [5].

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