  For an algebraic number $\alpha$ and $\gamma\in \mathbb{R}$, $h(\alpha)$ be
the (logarithmic) Weil height, and
$h_\gamma(\alpha)=(\mathrm{deg}\alpha)^\gamma h(\alpha)$ be the
$\gamma$-weighted (logarithmic) Weil height of $\alpha$. Let
$f:\overline{\mathbb{Q}}\to [0,\infty)$ be a function on the algebraic numbers
$\overline{\mathbb{Q}}$, and let $S\subset \overline{\mathbb{Q}}$. The
Northcott number $\mathcal{N}_f(S)$ of $S$, with respect to $f$, is the infimum
of all $X\geq 0$ such that $\{\alpha \in S; f(\alpha)< X\}$ is infinite. This
paper studies the set of Northcott numbers $\mathcal{N}_f(\mathcal{O})$ for
subrings of $\overline{\mathbb{Q}}$ for the house, the Weil height, and the
$\gamma$-weighted Weil height. We show:
  (1) Every $t\geq 1$ is the Northcott number of a ring of integers of a field
w.r.t. the house.
  (2) For each $t\geq 0$ there exists a field with Northcott number in $
[t,2t]$ w.r.t. the Weil height $h(\cdot)$.
  (3) For all $0\leq \gamma\leq 1$ and $\gamma'<\gamma$ there exists a field
$K$ with $\mathcal{N}_{h_{\gamma'}}(K)=0$ and
$\mathcal{N}_{h_\gamma}(K)=\infty$.
  For $(1)$ we provide examples that satisfy an analogue of Julia Robinon's
property (JR), examples that satisfy an analogue of Vidaux and Videla's
isolation property, and examples that satisfy neither of those. Item $(2)$
concerns a question raised by Vidaux and Videla due to its direct link with
decidability theory via the Julia Robinson number. Item (3) is a strong
generalisation of the known fact that there are fields that satisfy the Lehmer
conjecture but which are not Bogomolov in the sense of Bombieri and Zannier.
