3 × 72 × 127 × 337
1, 3, 7, 21, 49, 127, 147, 337, 381, 889, 1011, 2359, 2667, 6223, 7077, 16513, 18669, 42799, 49539, 128397, 299593, 898779, 2097151 and 6291453
No.
Its nearest Fibonacci number neighbors are 5702887 and 9227465.