22 × 3 × 11 × 17
1, 2, 3, 4, 6, 11, 12, 17, 22, 33, 34, 44, 51, 66, 68, 102, 132, 187, 204, 374, 561, 748, 1122 and 2244
No.
Its nearest Fibonacci number neighbors are 1597 and 2584.