Poulet numbers P for which P-q^2 is an abundant number for any q prime greater than 3

In this paper I make the following observation: there exist Poulet numbers P such that n = P – q^2 is an abundant number for any q prime, q ≥ 5 (of course, for q^2 < P). The first such P is 1105 (with corresponding [q, n] = [5, 1080], [7, 1056], [11, 984], [13, 936], [17, 816], [19, 744], [23, 576], [29, 264], [31, 144]). Another such Poulet numbers are 1387, 1729, 2047, 2701, 2821. Up to 2821, the Poulet numbers 341, 561, 645, 1905, 2465 don’t have this property. Questions: are there infinite many such Poulet numbers? What other sets of integers have this property beside Poulet numbers?