%I #53 Mar 22 2024 11:21:30

%S 0,0,0,1,2,2,1,2,3,2,2,4,3,1,3,2,2,5,3,0,4,3,2,5,5,1,4,3,1,5,3,2,6,3,

%T 0,6,5,2,6,6,0,6,5,1,6,5,1,4,3,0,7,5,2,5,6,2,9,7,1,8,6,0,6,4,0,8,5,1,

%U 3,7,2,9,7,0,7,5,2,9,6,0,9,5,0,7,11,1,6,6

%N Number of distinct greater twin primes which are in Goldbach partitions of 2n.

%C Tomas Oliveira e Silva in 2012 experimentally confirmed that all even numbers <= 4*10^18 have at least one Goldbach partition (GP) with a prime 9781 or less. Detailed examination of all even numbers < 10^6 showed that the most popular prime in all GPs is 3 (78497 occurrences), then 5 (70328), then 7 (62185), then 11 (48582), then 13 (40916), then 17 (31091), then 19 (29791) - all these primes are twin primes. These results gave rise to a hypothesis that twin primes should be rather frequent in GP, especially those relatively small.

%C Further empirical experiments demonstrated, surprisingly, there are in general two categories of even numbers n: category 1 - with 0, 1, or 2 distinct greater twin primes in all GPs(n), and category 2 - with fast increasing number of distinct greater twin primes in GPs(n).

%C Is a(n) = A294185(n-1)? - _R. J. Mathar_, Mar 22 2024

%H Marcin Barylski, <a href="/A294186/a294186.png">Plot of first 20000 elements of the A294186</a>

%H Marcin Barylski, <a href="/A294186/a294186_2.cpp.txt">C++ program for generating A294186</a>

%H Tomas Oliveira e Silva, <a href="http://sweet.ua.pt/tos/goldbach.html">Goldbach conjecture verification</a>

%e a(5)=2 because 2*5=10 has two ordered Goldbach partitions: 3+7 and 5+5. 5 is a greater twin prime (because 3 and 5 are twin primes), 7 is a greater twin prime (because 5 and 7 are twin primes).

%o (C++) See Barylski link.

%o (PARI) isgtwin(p) = isprime(p) && isprime(p-2);

%o a(n) = {vtp = []; forprime(p = 2, n, if (isprime(2*n-p), if (isgtwin(p), vtp = concat(vtp, p)); if (isgtwin(2*n-p), vtp = concat(vtp, 2*n-p)););); #Set(vtp);} \\ _Michel Marcus_, Mar 01 2018

%Y Cf. A002372 (number of ordered Goldbach partitions), A006512 (greater of twin primes), A294185, A295424.

%K nonn

%O 1,5

%A _Marcin Barylski_, Feb 11 2018