25 × 7 × 13
1, 2, 4, 7, 8, 13, 14, 16, 26, 28, 32, 52, 56, 91, 104, 112, 182, 208, 224, 364, 416, 728, 1456 and 2912
No.
Its nearest Fibonacci number neighbors are 2584 and 4181.