Polygon Numbers Associated with the Sum of Odd Divisors Function

2016-08-22T18:04:37Z (GMT) by Daeyeoul Kim Abdelmejid Bayad
<p>Let <i>n</i> be a positive integer. We investigate the sequences ((<i>S<sub>m</sub></i>(<i>n</i>))<sub><i>m</i></sub>, which concerns the iteration of the odd divisor function <i>S</i>, given by In this article, we introduced and studied the following invariants: <i>order, <i>m</i>-gonal shape number, type, convexity, and area of <i>n</i> derived from <i>S<sub>m</sub></i>(<i>n</i>).</i> For <i>m</i> = 3, 4, 5, we classify <i>m</i>-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 <i>m</i>-gonal shape numbers. An inductive algorithm for finding <i>m</i> + 1-gonal shape prime integers when an <i>m</i>-gonal shape integer is given. We give some examples and illustrations, for our conjectures and questions.</p>