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In a 2009 article, Barnett and Broughan considered the set of prime-index primes. If the prime numbers are listed in increasing order (2, 3, 5, 7, 11, 13, 17, . . .), then the prime-index primes are those which occur in a prime-numbered position in the list (3, 5, 11, 17, . . .). Barnett and Broughan established a prime-indexed prime number theorem analogous to the standard prime number theorem and gave an asymptotic for the size of the n-th prime-indexed prime.
We give explicit upper and lower bounds for π2(x), the number of prime-indexed primes up to x, as well as upper and lower bounds on the n-th prime-indexed prime, all improvements on the bounds from 2009. We also prove analogous results for higher iterates of the sequence of primes. We present empirical results on large gaps between prime-index primes, the sum of inverses of the prime-index primes, and an analog of Goldbach’s conjecture for prime-index primes.
Bayless, J., Klyve, D. & Oliveira e Silva, T. (2013). New bounds and computations on prime-indexed primes. Integers 13(A43), 1-21.