23 × 32 × 43
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 43, 72, 86, 129, 172, 258, 344, 387, 516, 774, 1032, 1548 and 3096
No.
Its nearest Fibonacci number neighbors are 2584 and 4181.