20181230, 13:34  #1 
May 2018
193 Posts 
Superprime gaps
Take the prime numbers at prime positions in the sequence of primes.
3, 5, 11, 17, 31, 41, 59, 67, 83, 109, ... What do we know about the gaps between primes in this sequence? 
20181230, 13:59  #2  
"Forget I exist"
Jul 2009
Dumbassville
20C0_{16} Posts 
Quote:
we can figure a gap of at least 6 after the first gap. and a gap of 12 or more due to gaps in primes of 4 or more. Last fiddled with by science_man_88 on 20181230 at 14:06 

20181230, 16:00  #3 
Aug 2006
1725_{16} Posts 
See also A073131.
Probably there are infinitely many gaps of length 6, but it seems hopeless to prove, even given a proof of the twin prime conjecture. The first few positions with such a gap: Code:
2, 3, 17, 405, 695, 891, 1016, 1406, 1782, 1886, 1982, 2052, 2070, 2078, 2753, 3131, 3758, 3949, 4130, 4133, 4312, 4561, 4745, 4922, 5307, 5415, 5462, 5917, 6457, 6925, 7022, 7459, 7802, 8268, 8923, 9025, 9265, 9787, 9849, 10119, 10522, 10962, 11153, 11299, 11678, 11958, 11962, 12087, 12109, 12129, 12317, 12396, 12753, 13335, 13685, 13804, 14062, 15369, 16148, 16314, 16888, 16921, 17092, 17112, 17154, 17271, 18251, 19726, 20282, 20572, 20863, 21030, 22580, 22753, 23913, 24479, 25379, 25476, 25845, 28051, 28125, 29811, 30818, 32257, 32837, 32960, 33030, 33067, 33085, 33295, 33312, 34167, 34229, 34524, 34583, 34850, 35088, 35502, 35932, 36636, 36827, 37133, 37281, 37909, 37950, 38239, 38528, 38709, 38782, 39331, 39419, 40253, 40399, 40739, 41804, 42375, 43089, 43180, 43432, 44236, 44529, 44568, 44801, 44828, 44960, 45283, 45327, 45394, 45633, 45787, 46269, 46327, 46559, 47008, 47235, 47668, 48038, 48766, 49835, 49892, 50051, 51352, 51476, 52206, 52347, 52554, 52971, 53043, 53617, 54725, 54934, 55000, 55074, 55170, 56372, 56390, 56887, 56929, 58683, 58707, 58752, 58950, 60612, 61393, 65527, 66647, 66684, 67330, 67387, 67941, 68111, 68732, 68765, 69281, 69305, 69352, 70035, 71172, 71573, 73506, 73826, 74460, 75259, 75275, 76092 
20190101, 18:18  #4 
May 2018
193 Posts 
These numbers are called primeindexed primes or superprimes. They are of the form p(p(n)), where p(n) is the nth prime number.
p(p(1))=p(2)=3 p(p(2))=p(3)=5 p(p(3))=p(5)=11 p(p(4))=p(7)=17 p(p(5))=p(11)=31 p(p(6))=p(13)=41 p(p(7))=p(17)=59 p(p(8))=p(19)=67 p(p(9))=p(23)=83 p(p(10))=p(29)=109 ... 
20190111, 14:47  #5 
May 2018
193 Posts 
The maximal gaps between superprimes are in A280080, A280081, and A280082 in OEIS. The first few maximal gaps are 2 between 3 and 5, 6 between 5 and 11, 14 between 17 and 31, 18 between 41 and 59, and 26 between 83 and 109. An interesting thing is that many maximal superprime gaps correspond to maximal prime gaps. There are a lot of maximal prime gaps starting with a prime n, where p(n) is the start of a maximal superprime gap. For example, p(2)=3, so the maximal prime gap between 2 and 3 corresponds to the maximal superprime gap between 3 and 5. Also, p(3)=5, so the prime gap between 3 and 5 corresponds to the superprime gap between 5 and 11. However, there are maximal superprime gaps not corresponding to a maximal prime gap. For example, the maximal superprime gap between 41 and 59 corresponds to the prime gap between 13 and 17, which is not maximal. It would be fun to see which maximal superprime gaps correspond to maximal prime gaps.

20190317, 20:01  #6 
May 2018
193 Posts 
Here are the superprimes under 1000.
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, ... Here are their gaps. 2, 6, 6, 14, 10, 18, 8, 16, 26, 18, 30, 22, 12, 20, 30, 36, 6, 48, 22, 14, 34, 30, 30, 48, 38, 16, 24, 12, 18, 92, 30, 34, 24, 62, 18, 42, 48, 24, ... 
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