[seqfan] new

[Tomasz Ordowski]

Dear readers!

Let Odd(m) be the odd part of m (the largest odd divisor of m).

By the dual Sierpinski conjecture;
if p is a prime, then there exists n > 0 such that Odd(p-1)+2^n is prime.
It seems that; if p is an odd prime p that is not a Fermat prime,
then Odd(p-1)+2^n is prime for infinitely many n.

By the dual Riesel conjecture;
if p is a prime, then there exists n such that |Odd(p+1)-2^n| is prime.
It seems that |Odd(p+1)-2^n| is prime for infinitely many n.

Let's define:
Primes p such that Odd(p+1)-2^n are composite for all 2^n < Odd(p+1).
3, 7, 31, 127, 661, 673, 1753, 1993, 2383, 2647, 2803, 2953, 3313, 3517,
3613, 3733, 3853, 3907, 6163, 6361, 6373, 7393, 7477, 7753, 7933, 8191, ...
These primes are not in the OEIS. They contain all Mersenne primes.

It should be noted that if the dual Riesel conjecture is true, then
these are primes p of the form k2^n-1, where k is a Polignac number.
Note that if k is a Riesel number, then such prime p does not exist.

For p = 3, 7, 31, 127, 661, 673, 1753, 1993, 2383, 2647, 2803, 2953, ...
Odd(p+1) = 1, 1, 1, 1, 331, 337, 877, 997, 149, 331, 701, 1477, ...
By the dual Riesel conjecture, it contains all de Polignac numbers
that are not Riesel numbers.

Best regards,

Thomas Ordowski
_______________________
https://en.wikipedia.org/wiki/Sierpinski_number#Dual_Sierpinski_problem
https://en.wikipedia.org/wiki/Riesel_number#The_dual_Riesel_problem
https://oeis.org/A006285 and https://oeis.org/A101036