23 × 5 × 7 × 13
1, 2, 4, 5, 7, 8, 10, 13, 14, 20, 26, 28, 35, 40, 52, 56, 65, 70, 91, 104, 130, 140, 182, 260, 280, 364, 455, 520, 728, 910, 1820 and 3640
No.
Its nearest Fibonacci number neighbors are 2584 and 4181.