3 × 7 × 31 × 37 × 79
1, 3, 7, 21, 31, 37, 79, 93, 111, 217, 237, 259, 553, 651, 777, 1147, 1659, 2449, 2923, 3441, 7347, 8029, 8769, 17143, 20461, 24087, 51429, 61383, 90613, 271839, 634291 and 1902873
No.
Its nearest Fibonacci number neighbors are 1346269 and 2178309.