Super-prime

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The subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime numbers begins

3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, … (sequence A006450 in OEIS).

That is, if p(i) denotes the ith prime number, the numbers in this sequence are those of the form p(p(i)); they have also been called super-prime numbers. Dressler & Parker (1975) used a computer-aided proof (based on calculations involving the subset sum problem) 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 (after the larger gap between super-primes 5 and 11) each super-prime number is less than twice its predecessor in the sequence.

One can also define "higher-order" primeness much the same way, and obtain analogous sequences of primes. Fernandez (1999)

A variation on this theme is the sequence of prime numbers with palindromic indices, beginning with

3, 5, 11, 17, 31, 547, 739, 877, 1087, 1153, 2081, 2381, … (sequence A124173 in OEIS).

[edit] References

Dressler, Robert E. & Parker, S. Thomas (1975), “Primes with a prime subscript”, Journal of the ACM 22 (3): 380–381, MR 0376599, DOI 10.1145/321892.321900 .
Fernandez, Neil (1999), An order of primeness, F(p), <http://borve.org/primeness/FOP.html> .