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All the terms are Fermat pseudoprimes to base 2 (A001567). For a proof see, e.g., Jaroma and Reddy (2007). - Amiram Eldar, Jul 24 2021

John H. Jaroma and Kamaliya N. Reddy, <a href="https://www.jstor.org/stable/27642303">Classical and alternative approaches to the Mersenne and Fermat numbers</a>, The American Mathematical Monthly, Vol. 114, No. 8 (2007), pp. 677-687.

Cf. A000215, A001567, A002808, A019434, A046052.

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From Giuseppe Coppoletta, May 06 2017: (Start)

Up to now it is not known if a non squarefree Fermat number could exist.

On the contrary, what is already known is that Fi is prime for i = 0..4 and composite for i = 5..32 (see Wagstaff ref). Moreover F5, F6, F7 and F8 have two prime factors, F9 has three prime factors, F10 has four, and F11 has five (see A046052).

These few cases could suggest a non-decreasing number of (distinct) prime factors of composite Fermat numbers. That, generally speacking, would be a quite strange property. Moreover, consider the following sequence, which can be seen as a Lucas-analogue of Fermat numbers: LF(n) = A000032(2^n) = A001566(n-1). LF verifies similar properties to Fermat numbers and, in particular, LF(n) is prime for n < 5 (apart LF(0) = 1), and after that it is composite, at least up to n = 20 (see A001606). Now LF(8) has 3 prime factors, LF(9) has 4, while LF(10) has 3 prime factors again (see Kelly's link). That could incline toward a similar negative answer for the monotone character of the number of prime factors of composite Fermat numbers.

Also, at my view and very loosely however, the primality patterns of sequences like LF considered above, somehow push a little bit further in the direction of "no more Fermat primes".

(End)

Blair Kelly, <a href="http://mersennus.net/fibonacci//lucas.txt">Factorizations of Lucas numbers</a>

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These few cases could suggest a non-decreasing number of (distinct) prime factors of composite Fermat numbers. That, generally speacking, would be a quite strange property. Moreover, consider the following sequence, which can be seen as a Lucas-analogue of Fermat numbers: LF(n) = A000032(2^n) = A001566(n-1). LF verifies similar properties to Fermat numbers and, in particular, LF(n) is prime for n < 5 (apart LF(0) = 1), and after that it is composite, at least up to n = 20 (see A001606). Now LF(8) has 3 prime factors, LF(9) has 4 and , while LF(10) has 3 prime factors again (see Kelly's link). That could incline toward a similar negative answer for the monotone character of the number of prime factors of composite Fermat numbers.

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