TITLE: Computable nonstandard models for weak systems of arithemtic
QUESTION [14 upvotes]: By Tennenbaum's theorem, PA itself does not have any computable nonstandard models.  The integer polynomials which are 0 or have a positive leading coefficient form a computable nonstandard model of Robinson arithmetic, which also happens to make the order relation total.  Since Presburger arithmetic is decidable, we can add axioms giving it a nonstandard number and work through Henkin's proof of the completeness theorem to get a computable nonstandard model of Presburger arithmetic.  (There's probably a simpler way to get one, though.)
Is any system strictly weaker than PA known to have no computable nonstandard models?
What other systems weaker than PA are known to have computable nonstandard models?
.
possible examples of either include:
I-Delta-0, I-Delta-0(exp), I-Sigma-1
Elementary Function Arithmetic
Elementary Recursive Arithmetic, Primitive Recursive Arithmetic
Robinson arithmetic + Euclidean division, Robinson arithmetic + Euclidean division + order relation is total

REPLY [6 votes]: $\mathsf{IE_1}$ doesn't have computable non-standard models. (George Wilmers, "Bounded existential induction", 1985) Any theory that contains it will not have computable non-standard models, e.g.: $\mathsf{I\Delta_0}$, $\mathsf{PRA}$, ...
On the other hand, $\mathsf{IOpen}$ does have computable non-standard models (J. C. Shepherdson, "A non-standard model for a free variable fragment of number theory", 1964). Any theory contained in it will also have computable non-standard models, e.g.: $\mathsf{Q}$.
The threshold(s) of having a computable non-standard model is somewhere between $\mathsf{IOpen}$ and $\mathsf{IE_1}$. There are various principles that one can add to $\mathsf{IOpen}$ and see if the resulting theory still has a computable non-standard model. 
It is known that $\mathsf{IOpen}$ plus cofinality of primes and Bezout axioms has computable non-standard models. Over $\mathsf{IOpen}$, Bezout implies normality and GCD axioms and is provable in $\mathsf{IE_1}$.