All the simple examples of undecidable problems that I know deal with symbolic computation or calculation. For example, the halting problem, whether Diophantine equations have solutions, the word problem for groups, the Post correspondence problem. The post An example of an easy to understand undecidable problem provides some other examples.
Any of these can be turned into an undecidable decision problem for a subset of $\Bbb N$ by suitably encoding the problem instances as numbers.
I would like an example of undecidable subset of $\Bbb N$ that is presented directly, defined in terms of some predicate $\phi$ that applies as directly as possible to the arithmetic properties of the numbers themselves
rather than something like “$\phi(n)$ holds if $n$ is the index of a Turing machine $\#n$ for which…” or “$\phi(n)$ means that $n$, considered as representing a Lisp program encoded in ASCII…”.
Is there such an example?
(This question is similar in spirit to the one that asks if there is an undecidable theorem of arithmetic that isn't of the highly artificial type that is constructed by Gödel's incompleteness theorem. There the usual examples are Goodstein's theorem or the Parris-Harrington theorem.)