Polygon Numbers Associated with the Sum of Odd Divisors Function

Let n be a positive integer. We investigate the sequences ((S_m(n))_m, which concerns the iteration of the odd divisor function S, given by (Formula presented.) In this article, we introduced and studied the following invariants: order, m-gonal shape number, type, convexity, and area of n derived from S_m(n). For m = 3, 4, 5, we classify m-gonal shape odd prime integers. According to some numerical computational evidence in the Appendix (see supplemental data), we state conjectures and questions with respect to the existence problem for m-gonal shape numbers. An inductive algorithm for finding m + 1-gonal shape prime integers when an m-gonal shape integer is given. We give some examples and illustrations, for our conjectures and questions.