Sierpiński Number of the First Kind

A Sierpiński number of the first kind is a positive integer of the form , where is a positive integer. The first few are 2, 5, 28, 257, 3126, 46657, 823544, 16777217, ... (OEIS A014566). Sierpiński proved that if is prime with , then must be of the form , making a Fermat number with for a nonnegative integer . The first few of this form are 1, 3, 6, 11, 20, 37, 70, ... (OEIS A006127).

For a nonnegative integer , the number of digits in the candidate is given by

where is the ceiling function, so the numbers of digits in the first few candidates are 1, 3, 20, 617, 315653, 41373247568, ... (OEIS A089943).

The only known prime Sierpiński numbers of the first kind are 2, 5, 257, with the first unknown case being . The status of Sierpiński numbers is summarized in the table below (Nielsen).

		status of
0	1	prime ()
1	3	prime ()
2	6	composite with factor
3	11	composite with factor
4	20	composite with no factor known
5	37	composite with factor
6	70	unknown
7	135	unknown
8	264	unknown
9	521	unknown
10	1034	unknown
11	2059	composite with factor
12	4108	unknown
13	8205	unknown
14	16398	unknown
15	32783	unknown
16	65552	unknown
17	131089	unknown