20190802, 21:33  #45  
Jul 2019
7_{10} Posts 
Quote:
In this context, we could talk about gaps that are fun or not fun. For example, memory gaps are not fun. However, prime gaps are fun. Indeed, for Bobby Jacobs, prime gaps are very fun. Yes, offtopic, and probably pedantic, but this discussion was also fun. A little more ontopic, I do consider prime gaps to be quite "fun". I can remember as early as the 1990's reading about the race to discover bigger and bigger primes, especially Mersenne Primes. The primes themselves were the flashy new thing that people were interested in. But now, I'm more interested in the prime numbers as a whole, as a set. I'm interested in the "structure", and how that structure seems to blend determinism and randomness. I was shocked to learn just a few years ago that prime gaps of length 6 are far more common than gaps of length 2 or 4. I understood that the "density" of primes is approximately the reciprocal of the logarithm of the numbers being considered, but I had never really given much thought to the structure of the set of primes. I was further shocked to learn that the number of primes so closely approximates the logarithmic integral. I mean, it makes sense that if the density is approximately the reciprocal of the logarithm, then the total number of primes should be approximately the integral of that function, i.e., the logarithmic integral function. But in my mind, the word "approximately" implied quite a bit of potential variance. It was only after reading something about a year ago that it finally sank it. If the prime counting function has an error term of O(log(x) sqrt(x)), then when x is large enough, the logarithmic integral is extremely accurate. For example, using only the logarithmic integral, we can calculate the number of primes less than a googol (i.e., 10^100) to about 46 digits of accuracy. That shocked me to learn. I had no idea that it was that precise. I've been interested in the Riemann Hypothesis for a couple decades, but it's amazing to me how little I really know about prime numbers. For me to know so little about primes, I often wonder if I should bother caring about the RH. So I study the primes, the structure. I want to learn more, run experiments, get an intuitive feel for the primes. And I want to learn some of the theory, but in a practical, handson way, so that I really understand it. Years ago I had this naive daydream of someday proving (or disproving) the RH, but now I just want to learn more about primes, and maybe contribute some small bit to our collective understanding. 

20190803, 21:01  #46  
Jun 2015
Vallejo, CA/.
5×197 Posts 
Quote:
Quote:
However, there is always a question I've had. Which number do we take for a gap of size 6? Ordinarily, it would be the sequence http://oeis.org/A023201, because when defining "sexy" primes it doesn't matter if there is a prime (or not) between them. While the gaps of 2 and 4 are (with one exception) always contiguous, the gaps of 6 or 30 may or may not be like that. Take for instance gaps of 6 up to 100 5,11 7,13 11,17 13,19 17,23 23,29 31,37 41,47 47,53 53,59 61,67 67,73 73,79 83,89 97,103 see http://oeis.org/A023201 Of all of these only 23,29 31,37 47,53 53,59 61,67 73,79 83,89 are consecutive primes see http://oeis.org/A031924 As the numbers grow bigger both sequences (A023201 and A031924) start to look alike. The only "sexy" primes that have a prime between them are the TRIPLETS. The density of triplets is one order of magnitude less than that of the sexy primes. Between 23 and 554,893 there are 8,158 pairs of primes with gap 6 Between 5 and 554,893 there are 10,000 pairs of "sexy primes". between 5 and 554,887 there 1842 set of triplets (either { p, p+2, p+6} or {p, p+4, p+6} The density of Prime triplet becomes sparse for higher numbers, below 100 million there are only 111,156 triplets. (1/10 of 1%) So for sufficiently high intervals the number of Sexy primes approximates the numbers of primes with gap 6 So to compare twin primes, sexy primes, cousin primes and triplets Code:
SEXY PRIMES TWIN PRIMES COUSIN PRIMES TRIPLETS Under 1,299,709 UNDER 1,299,709 UNDER 1,299,709 UNDER 1,299,709 24,168 20,498 16,943 3,483 So even if we discard the "sexy" that have a prime in between we still have more primes with gap 6 than twins 20685 vs 20498 This numbers (gap 2 vs gap 6) will not remain close, because as the tested interval go higher triplets become sparser and sparser. If I have time i'll try to see how many "sexy" primes there are under 10 million and compare it with "twin" primes, "cousin" primes and "triplets". 

20190803, 22:23  #47 
Jun 2015
Vallejo, CA/.
5×197 Posts 
They were some mistakes in the charts as I used info from different databases.
This is corrected version of the table So to compare twin primes, sexy primes, cousin primes and triplets all under 1'040,000 Code:
SEXY PRIMES TWIN PRIMES COUSIN PRIMES TRIPLETS Under 1,04000 UNDER 1,040,000 UNDER 1,040,000 UNDER 1,040,000 16,951 8,464 8,438 2,935 https://primenumbers.info/#numberTypes As can be seen regardless of how we count gaps of 6 –either pure gaps like {23,29} alone or if we include gaps that belong to a triplet like {103,109}– the fact remains that primes with a gap of 6 are (at least in these ranges) almost twice than twin primes. Last fiddled with by robert44444uk on 20190805 at 11:18 Reason: Poster out of time to make change 
20191223, 22:46  #48 
May 2018
326_{8} Posts 
I was on OEIS recently and saw that some of the prime gap sequences are now on OEIS. The gaps between record gaps between record prime gaps are A326747, and the record gaps between record gaps between record prime gaps are A326829.

20191224, 10:04  #49 
Jun 2003
Oxford, UK
1,933 Posts 
Interesting, but as Mr Sloane says on A326747, "let us stop here"

20200708, 21:06  #50 
May 2018
326_{8} Posts 
Unfortunately, the sequences of gaps between record gaps between record prime gaps and record gaps between record gaps between record prime gaps are no longer on OEIS anymore. The sequences A326747, A326829 are now different sequences.

20200708, 21:27  #51  
Sep 2002
Database er0rr
2·11·167 Posts 
Quote:


20200709, 07:49  #52  
"Jeppe"
Jan 2016
Denmark
2^{3}×3×7 Posts 
Quote:
A326747 revision #56 A326829 revision #34 /JeppeSN 

20200822, 15:20  #53  
May 2018
2×107 Posts 
Quote:


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