3 × 72 × 43 × 127
1, 3, 7, 21, 43, 49, 127, 129, 147, 301, 381, 889, 903, 2107, 2667, 5461, 6223, 6321, 16383, 18669, 38227, 114681, 267589 and 802767
No.
Its nearest Fibonacci number neighbors are 514229 and 832040.