$a_1=k,a_{n}=2a_{n-1}+1(ngeq 2).$ Does there exist $kinmathbb N$ such that $a_n,n=1,2,3,cdots$ are all composite numbers?

Let $a_1=k,a_{n}=2a_{n-1}+1(n\geq 2).$

If $k=1$ then $a_n=1,3,7,15,31,63,\cdots$ here $3,7,31$ are prime numbers. I'm interested in this problem:

Does there exist $k\in\mathbb N$ such that $a_n,n=1,2,3,\cdots$ are all composite numbers?

If $k=147$ then $a_n,n=1,2,\cdots 2551$ are all composite, but $a_{2552}$ is prime. So I doubt the existence of such number.

Answer

The numbers you mention are Riesel numbers http://en.wikipedia.org/wiki/Riesel_number
and there is the similar Sierpinski numbers where it is $2a_{n-1}-1$ instead.

http://en.wikipedia.org/wiki/Sierpinski_number