25 × 3 × 7
1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 32, 42, 48, 56, 84, 96, 112, 168, 224, 336 and 672
No.
Its nearest Fibonacci number neighbors are 610 and 987.