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.