[seqfan] A245730 and new research

[Tomasz Ordowski]

Dear SeqFans!

Let's define:
Odd primes p = k2^m+1 = k+2^n with k odd.
These are odd primes p such that p - Odd(p-1) = 2^n,
where Odd(p-1) is the largest odd divisor (odd part) of p-1.
Question: How to prove that A245730 are the same primes?
See https://oeis.org/A245730 (especially the second comment).

A slightly different definition:
Primes p = k2^m-1 = k+2^n with k odd.
It gives similar primes: 3, 5, 11, 17, 43, 257,
683, 2731, 43691, 65537, 174763, 2796203, ...
These are primes p such that p - Odd(p+1) = 2^n.
It includes all Fermat primes and all Wagstaff primes:
https://oeis.org/A019434 and https://oeis.org/A000979
Are there no other primes among them? Let's find out!

Best regards,

Thomas Ordowski
________________
Still to be explored:
(*) Primes p = k2^m+1 = 2^n-k with k odd.
These are primes p such that p+Odd(p-1) = 2^n.
(**) Primes p = k2^m-1 = 2^n-k with k odd.
These are primes p such that p+Odd(p+1) = 2^n.