Asymptotics about A005228

David W. Wilson wilson.d at
Tue Jun 3 15:14:27 CEST 2008

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,


    [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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