Problems & Puzzles: Puzzles

Puzzle 312. Sequence of primes such that...

Frank Rubin sent the following puzzle:

It is possible to write an infinite sequence of primes, such that the sum of any three consecutive terms is also prime.  One such sequence begins
   3, 5, 11, 7, 13, 17, 29, 43, 41, ...
The corresponding sequence of sums begins
   19, 23, 31, 37, 59, 89, 113, ...
Q1. Is possible to find such a sequence where every odd prime appears exactly once in either the original sequence or the sequence of sums?
A sequence can be formed using the following rule:  Let the first two primes be 3 and 5.  After that, let the next prime P(i) be the first prime that has not appeared yet, for which P(i-2)+P(i-1)+P(i) is prime.

Q2. Does this rule produce the desired sequence where every odd prime appears exactly once?

I have not looked, but it seems likely that someone has already investigated this.  For all I know, this problem is already on your website.



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