[seqfan] Two problems on primes

[Vladimir Shevelev]

Dear SeqFans,
 
In my opinion, the following two questions are of interest.
 
Let p an odd prime such that 2^n<p<2^(n+1). Consider a finite sequence B_p={b_i=2^i*(p+1)-1, i=1,...,n}. 
Question 1. Are p=3 and p=5 only primes for which all terms of B_p are  primes?
Note that, if p>=7, then, for p==7(mod10) , b_1==0(mod5);  for p==3(mod10), b_2==0(mod5), for p==1(mod10), b_3==0(mod5). Thus in nontrivial case p==9(mod10).
For example, let p=89. Then n=6. Here the first 5 terms of B_p are primes: b_1=179, b_2=359, b_3=719, b_4=1439, b_5=2879 and only b_6=5759=13*443.
Consider now sequence of records R={r(p), p=3,5,...} of the number of the first prime terms in B_p. So, the first 3 records are r(3)=1, r(5)=2, r(89)=5,...
Can anyone to continue this sequence?
Question 2. Is sequence R infinite?
 
Regards,
Vladimir

 Shevelev Vladimir‎