Question 149
Let $\mathbb{N}$ be the set of positive integers. Define
$$
A = \{2^a + a^2 \mid a \in \mathbb{N}\}
$$
and, for each $i \in \mathbb{N}$,
$$
h_1(n) = n + \log_2 n; \quad h_{i+1}(n) = n + \log_2 h_i(n), \quad \forall i \in \mathbb{N}.
$$
Show that there exists a one-to-one function $g : \mathbb{R}^+ \to \mathbb{R}^+$ such that
(a) For $i, n \in \mathbb{N}$, there holds $n + \lfloor g(h_i(n)) \rfloor \notin A$.
(b) For each $k \in \mathbb{N} \setminus A$ and $i \in \mathbb{N}$, there uniquely exists $a_{k,i} \in \mathbb{N}$ satisfying
$$
a_{k,i} + \lfloor g(h_i(a_{k,i})) \rfloor = k, \quad \forall i \in \mathbb{N}.
$$
Here $\lfloor x \rfloor$ is the greatest integer that does not exceed the real number $x$.
$$
A = \{2^a + a^2 \mid a \in \mathbb{N}\}
$$
and, for each $i \in \mathbb{N}$,
$$
h_1(n) = n + \log_2 n; \quad h_{i+1}(n) = n + \log_2 h_i(n), \quad \forall i \in \mathbb{N}.
$$
Show that there exists a one-to-one function $g : \mathbb{R}^+ \to \mathbb{R}^+$ such that
(a) For $i, n \in \mathbb{N}$, there holds $n + \lfloor g(h_i(n)) \rfloor \notin A$.
(b) For each $k \in \mathbb{N} \setminus A$ and $i \in \mathbb{N}$, there uniquely exists $a_{k,i} \in \mathbb{N}$ satisfying
$$
a_{k,i} + \lfloor g(h_i(a_{k,i})) \rfloor = k, \quad \forall i \in \mathbb{N}.
$$
Here $\lfloor x \rfloor$ is the greatest integer that does not exceed the real number $x$.