23 × 3 × 29 × 59
1, 2, 3, 4, 6, 8, 12, 24, 29, 58, 59, 87, 116, 118, 174, 177, 232, 236, 348, 354, 472, 696, 708, 1416, 1711, 3422, 5133, 6844, 10266, 13688, 20532 and 41064
No.
Its nearest Fibonacci number neighbors are 28657 and 46368.