Prime Sums

[hv at crypt.org]

"Eric Angelini" <keynews.tv at skynet.be> wrote:
:
: Hello SeqFan,
: would those 5 new seq. have their place in the new OEIS?
: Best,
: É.
: http://www.cetteadressecomportecinquantesignes.com/P1P2P3P4.htm

The description says:
  Start with the first Prime > 2 not used so far [3]; call it P_1;
  find the smallest unused prime P_2 so to build the equality
  P_1 + P_2 = P_3 + P_4 (here: 3 + 13 = 5 + 11). Repeat.

I think the description here is incomplete in a couple of respects.
My guess is that a) P_3 and P_4 must also be not-yet-used primes,
and b) P_3 is chosen to be the smallest permissible prime once P_2
has been fixed.

Having these as 5 separate sequences will make it difficult to see
the overview: I'd be tempted to have a sequence P_n either as well as
or instead of the separate P_1..P_4 sequences:

3, 13, 5, 11, 7, 29, 17, 19, ...

 From time to time, this sequence will have no gaps in the primes
used so far; the primes at which this occurs may also be a sequence
of interest:

3, 13, 29, 61, 79, 139, ...

Is it easy to show that this last sequence is infinite?

Hugo