[seqfan] Extensions of Mersenne exponents to composite numbers

[Tomasz Ordowski]

Dear SeqFans!

As is well known, if the Mersenne number M_p = 2^p - 1 is prime, then p
must be a prime.
Note that, by Fermat Little Theorem, for a prime p, we have M_p = 2^p - 1
== 1 (mod p).
I proposed the following generalization of Mersenne exponents on natural
numbers:
(*) Numbers n such that q = 2^n - (2^n mod n) + 1 is prime. See A270427.
Other extensions that maintain the congruence q == 1 (mod n):
(**) Numbers n indivisible by 4 such that q = 2^(phi(n)+1) - 1 is a
Mersenne prime.
(***) Numbers n such that q = 2^n - 2^m + 1 is prime, where m = A270096(n).
Or, using the Redei theorem b^n == b^(n-phi(n)) (mod n), I suggest:
(****) Numbers n such that q = 2^n - 2^(n-phi(n)) + 1 is prime.

I am asking for comments.

Best regards,

Thomas Ordowski