Superprime gaps

Bobby Jacobs · 2018-12-30, 13:34

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?

science_man_88 · 2018-12-30, 13:59

Quote:

Originally Posted by Bobby Jacobs

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?

http://oeis.org/A006450

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.

CRGreathouse · 2018-12-30, 16:00

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

Bobby Jacobs · 2019-01-01, 18:18

These numbers are called prime-indexed 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
...

Bobby Jacobs · 2019-01-11, 14:47

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.

Bobby Jacobs · 2019-03-17, 20:01

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, ...

2018-12-30, 13:34	#1
Bobby Jacobs May 2018 102₁₆ 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?

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2019-01-01, 18:18	#4
Bobby Jacobs May 2018 2×3×43 Posts	These numbers are called prime-indexed 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 ...

2019-01-11, 14:47	#5
Bobby Jacobs May 2018 2·3·43 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.

2019-03-17, 20:01	#6
Bobby Jacobs May 2018 258₁₀ 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, ...