[seqfan] Two problems and two questions

[Tomasz Ordowski]

Hello all,

I have new problems and questions.

   There are odd numbers k>1 such that
(k-2^m)2^n-1 is composite for every 2^m<k and 2^n<k,
namely 65535 and 13975275 [found by Amiram Eldar].
Problem 1. Are there infinitely many such numbers?
Question 1. Are there odd numbers k>1 such that
(k-2^m)2^n+1 is composite for every 2^m<k and 2^n<k ?
Note that such a number k is composite (put m=n=0).

   There are odd numbers k>1 such that
(k+2^m)2^n+1 is composite for every 2^m<k and 2^n<k,
namely 14568915 [Amiram Eldar found only this one].
Problem 2. Are there infinitely many such numbers?
Question 2. Are there odd numbers k>1 such that
(k+2^m)2^n-1 is composite for every 2^m<k and 2^n<k ?
Note that such a number k is composite (put m=n=0).

Maybe someone will find answers to these questions.

Best regards,

Thomas Ordowski
____________________
Cf. https://oeis.org/draft/A337487
https://oeis.org/history/view?seq=A337487&v=16
Odd integers k>3 that are not of the form prime+2^m+2^n.