Super-prime

Super-prime numbers, also known as higher-order primes or prime-indexed primes (PIPs), 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, the primes matched with prime ordinal numbers are the super primes.

The subsequence begins
 * 3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, 179, 191, 211, 241, 277, 283, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991, ....

That is, if p(n) denotes the nth prime number, the numbers in this sequence are those of the form p(p(n)).

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.

show that there are
 * $$\frac{x}{(\log x)^2} + O\left(\frac{x\log\log x}{(\log x)^3}\right)$$

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
 * 3, 5, 11, 17, 31, 547, 739, 877, 1087, 1153, 2081, 2381, ....