Geodesic
On the subsequence of primes having prime subscripts
Journal of integer sequences, Tome 12 (2009) no. 2
We explore the subsequence of primes with prime subscripts, $(q_{n})$, and derive its density and estimates for its counting function. We obtain bounds for the weighted gaps between elements of the subsequence and show that for every positive integer $m$ there is an integer arithmetic progression $(an+b : n \in \Bbb N)$ with at least $m$ of the $(q_{n})$ satisfying $q_{n} = an+b$.
Classification : 11A41, 11B05, 11B25, 11B83
Keywords: prime-prime, prime-prime number theorem, prime-prime gaps, prime-primes in progressions 
@article{JIS_2009__12_2_a2,
     author = {Broughan,  Kevin A. and Barnett,  A.Ross},
     title = {On the subsequence of primes having prime subscripts},
     journal = {Journal of integer sequences},
     year = {2009},
     volume = {12},
     number = {2},
     zbl = {1228.11010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2009__12_2_a2/}
}
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Broughan,  Kevin A.; Barnett,  A.Ross. On the subsequence of primes having prime subscripts. Journal of integer sequences, Tome 12 (2009) no. 2. http://geodesic.mathdoc.fr/item/JIS_2009__12_2_a2/