For an algebraic number (Formula presented.) and (Formula presented.), let (Formula presented.) be the house, (Formula presented.) be the (logarithmic) Weil height, and (Formula presented.) be the (Formula presented.) -weighted (logarithmic) Weil height of (Formula presented.). Let (Formula presented.) be a function on the algebraic numbers (Formula presented.), and let (Formula presented.). The Northcott number (Formula presented.) of (Formula presented.), with respect to (Formula presented.), is the infimum of all (Formula presented.) such that (Formula presented.) is infinite. This paper studies the set of Northcott numbers (Formula presented.) for subrings of (Formula presented.) for the house, the Weil height, and the (Formula presented.) -weighted Weil height. We show: (1)Every (Formula presented.) is the Northcott number of a ring of integers of a field w.r.t. the house (Formula presented.). (2)For each (Formula presented.), there exists a field with Northcott number in (Formula presented.) w.r.t. the Weil height (Formula presented.). (3)For all (Formula presented.) and (Formula presented.), there exists a field (Formula presented.) with (Formula presented.) and (Formula presented.). For (1) we provide examples that satisfy an analogue of Julia Robinson'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.