3 × 17 × 257 × 65537
1, 3, 17, 51, 257, 771, 4369, 13107, 65537, 196611, 1114129, 3342387, 16843009, 50529027, 286331153 and 858993459
No.
Its nearest Fibonacci number neighbors are 701408733 and 1134903170.