Super-prime

Super-prime numbers, also known as higher-order primes or prime-indexed primes, are the subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime numbers. In other words, if prime numbers are matched with ordinal numbers, starting with prime number 2 matched with ordinal number 1, then the primes matched with prime ordinal numbers are the super-primes.
The subsequence begins
That is, if p denotes the nth prime number, the numbers in this sequence are those of the form p.

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
p	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71
p	3	5	11	17	31	41	59	67	83	109	127	157	179	191	211	241	277	283	331	353

In 1975, Robert Dressler and Thomas Parker used a computer-aided proof to show that every integer greater than 96 may be represented as a sum of distinct super-prime numbers. Their proof relies on a result resembling Bertrand's postulate, stating that each super-prime number is less than twice its predecessor in the sequence.
A 2009 research showed that there are
super-primes up to x.
This can be used to show that the set of all super-primes is small.
One can also define "higher-order" primeness much the same way and obtain analogous sequences of primes.
A variation on this theme is the sequence of prime numbers with palindromic prime indices, beginning with

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
p	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71
p	3	5	11	17	31	41	59	67	83	109	127	157	179	191	211	241	277	283	331	353

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
p	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71
p	3	5	11	17	31	41	59	67	83	109	127	157	179	191	211	241	277	283	331	353

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
p	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71
p	3	5	11	17	31	41	59	67	83	109	127	157	179	191	211	241	277	283	331	353