As far as nonstandard models go: we can indeed get -like intervals such that each has a standard factor. The proof is via Compactness, and the Chinese Remainder Theorem:
First, adjoin a constant symbol to our language. Let be the prime number, let , and let .
Define numbers , by recursion as follows:
,
,
Now, for each , let express " is congruent to (mod)", let express " is congruent to (mod)," and let . By the Chinese Remainder Theorem, every finite subset of is consistent with True Arithmetic , so by Compactness, itself is consistent with . So there is some nonstandard model of in which holds; clearly, in such a model, every number in the -like interval centered on has a standard factor.
I have no idea whether nonstandard model has such an interval, however.
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