This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A265684 #20 Jun 29 2019 02:24:48
%S A265684 645,7957,11305,15841,25761,35333,126217,194221,212421,332949,464185,
%T A265684 635401,656601,741751,934021,1193221,1357441,1459927,1620385,1690501,
%U A265684 1969417,2704801,3911197,4154161,4209661,5095177,5284333,5351537,5758273,6189121,6212361,7820201,8134561,8209657
%N A265684 Sarrus numbers (A001567) that are the average of two consecutive primes.
%C A265684 Inspired by A265669.
%C A265684 Motivation was the form of differences between consecutive primes that generate this sequence. It seems that 12*k appears in differences most of the time. For the first 175 term of this sequence, the relevant proportion is 161/175.
%C A265684 Differences between corresponding consecutive primes are 4, 12, 12, 36, 4, 12, 12, 36, 4, 4, 24, 24, 4, 60, 24, 24, 24, 12, 12, 36, 12, 24, 12, 24, 36, 12, 12, 12, 12, 24, 4, 60, 24, 48, 36, 12, 24, 36, 24, 20, 12, 84, 36, 12, 24, 24, 12, 24, 36, 12, 12, 36, ...
%H A265684 Amiram Eldar, <a href="/A265684/b265684.txt">Table of n, a(n) for n = 1..10000</a>
%e A265684 645 is a term because it is a Sarrus number and the average of the consecutive primes 643 and 647.
%e A265684 7957 is a term because it is a Sarrus number and the average of the consecutive primes 7951 and 7963.
%t A265684 Select[Range[200000], CompositeQ[#] && PowerMod[2, (# - 1), #] == 1 && NextPrime[#] - # == # - NextPrime[#, -1] &] (* _Amiram Eldar_, Jun 28 2019 *)
%o A265684 (PARI) is(n)={Mod(2, n)^n==2 && !isprime(n)}
%o A265684 forcomposite(n=2, 1e7, if(is(n) && (nextprime(n)-n)==(n-precprime(n)), print1(n, ", ")))
%Y A265684 Intersection of A001567 and A024675.
%Y A265684 Cf. A265669.
%K A265684 nonn
%O A265684 1,1
%A A265684 _Altug Alkan_, Dec 13 2015