32 × 7 × 43 × 127
1, 3, 7, 9, 21, 43, 63, 127, 129, 301, 381, 387, 889, 903, 1143, 2667, 2709, 5461, 8001, 16383, 38227, 49149, 114681 and 344043
No.
Its nearest Fibonacci number neighbors are 317811 and 514229.