Just an odd numbers, which are produced by repeatedly adding
1 (or invoke a
successor function) such that the resulting pile cannot be divided into
n of equal parts without a reminder.
Just this. There is no deep mystery whatsoever. Some piles or aggregates cannot be divided into equal parts without leaving a reminder.
By the way, Mother Nature never use this fact since it could not "see" it. Mother Nature uses pattern-matching on "concrete" molecular (or atomic if you wish) structures with explicit start and end marks (or primers).