3 × 7 × 37 × 333667
1, 3, 7, 21, 37, 111, 259, 777, 333667, 1001001, 2335669, 7007007, 12345679, 37037037, 86419753 and 259259259
No.
Its nearest Fibonacci number neighbors are 165580141 and 267914296.