In this experiment, I have checked how many times different gapped primes occur out of the first 10000, 100000, 1000000 first primes.


Please view the following as ($X$:$Y$) where $X$ represents the gap and $Y$ represents how many times it occurs.


Out of first 1000000 primes:


**2:** 40405
**4:** 40233
**6:** 68311
**8:** 28746
**10:** 36766
**12:** 44113
**14:** 23569
**16:** 16658
**18:** 29065
**20:** 14409
**22:** 12833
**24:** 17680
**26:** 7979
**28:** 8493
**30:** 13773
**32:** 4048
**34:** 4263
**36:** 6344
**38:** 2749
**40:** 3313
**42:** 4424
**44:** 1717
**46:** 1387
**48:** 2235


Out of first 100000 primes:


**2:** 10251
**4:** 10213
**6:** 15989
**8:** 7067
**10:** 8873
**12:** 10158
**14:** 5353
**16:** 3661
**18:** 6304
**20:** 3043
**22:** 2826
**24:** 3538
**26:** 1543
**28:** 1631
**30:** 2114
**32:** 742
**34:** 756
**36:** 1032
**38:** 455
**40:** 563
**42:** 661
**44:** 250
**46:** 219
**48:** 290


Out of first 10000 primes:


**2:** 1271
**4:** 1263
**6:** 2012
**8:** 801
**10:** 953
**12:** 1008
**14:** 512
**16:** 353
**18:** 537
**20:** 249
**22:** 235
**24:** ///22///
**26:** 91
**28:** 102
**30:** 154
**32:** 35
**34:** 36
**36:** 55
**38:** 20
**40:** 28
**42:** ///20///
**44:** 5
**46:** 6
**48:** ///3///


You will notice the following:


* A peak occurs in almost all $X \mod 6 =0$ compared to the prior 2. The only exceptions are in the first first 10000 primes and only in $X=24$, $X=42$ and $X=48$
* There are always more shorter gaps than longer gaps


My question is broken into 3 parts:


A) Is it fair to assume that probably there are more shorter gaps than longer gaps?


B) Is it fair to assume that probably there are more $X \mod 6=0$ gaps than any other gaps?


C) If B) is assumed to be fair, is it fair to assume that SEXY primes (primes with a gap of 6) are the most popular gap between primes?



Comparing densities of different gapped primes (twin, cousin, sexy...)
Andrew Odlyzko, Michael Rubinstein, and Marek Wolf, Jumping champions, Experimental Mathematics 8 (1999), 107–118 suggest that somewhere around $x=1.7427\times10^{35}$, the most common gap between consecutive primes less than $x$ switches from $6$ to $30$.


See The Most Common Prime Gaps, posted by John Baez, at <https://golem.ph.utexas.edu/category/2016/03/the_most_common_prime_gaps.html>


