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A295424 Number of distinct twin primes which are in Goldbach partitions of 2n.

%I A295424 #38 Aug 06 2018 05:31:27
%S A295424 0,0,1,2,3,2,3,4,4,4,5,6,4,3,5,4,6,7,3,4,6,5,6,9,6,4,7,4,5,8,5,7,8,3,
%T A295424 6,10,7,7,11,6,6,10,6,6,11,6,4,7,3,7,11,7,6,10,8,10,15,8,8,14,6,6,10,
%U A295424 4,8,12,6,3,10,9,10,15,7,7,12,7,10,14,6,9,13,5,7
%N A295424 Number of distinct twin primes which are in Goldbach partitions of 2n.
%C A295424 Tomas Oliveira e Silva in 2012 experimentally confirmed that all even numbers 4 <= n <= 4 * 10^18 have at least one Goldbach partition (GP) with a prime 9781 or less. Detailed examination of all even numbers less than 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 A295424 Conjecture. Further empirical examinations lead to a hypothesis that all even numbers n > 4 have at least 1 twin prime in GP(n).
%C A295424 a(n) <= A294185(n) + A294186(n).
%H A295424 Marcin Barylski, <a href="/A295424/a295424.png">Plot of first 20000 elements of the A295424</a>
%H A295424 Marcin Barylski, <a href="/A295424/a295424_1.cpp.txt">C++ program for generating A295424</a>
%H A295424 Marcin Barylski, <a href="http://tas-moto.org/research/TwinPrimesInGoldbachPartitions.pdf">Studies on Twin Primes in Goldbach Partitions of Even Numbers</a>
%H A295424 Tomas Oliveira e Silva, <a href="http://sweet.ua.pt/tos/goldbach.html">Goldbach conjecture verification</a>
%e A295424 a(5) = 3 because 5 * 2 = 10 has 2 ordered Goldbach partitions: 3 + 7 and 5 + 5 and primes 3, 5, 7 are distinct twin primes in this set.
%o A295424 (C++) // See Barylski link.
%o A295424 (PARI) istwin(p) = isprime(p) && (isprime(p-2) || isprime(p+2));
%o A295424 a(n) = {vtp = []; forprime(p= 2, n, if (isprime(2*n-p), if (istwin(p), vtp = concat(vtp, p)); if (istwin(2*n-p), vtp = concat(vtp, 2*n-p)););); #Set(vtp);} \\ _Michel Marcus_, Mar 01 2018
%Y A295424 Cf. A294186, A294185, A001097.
%K A295424 nonn
%O A295424 1,4
%A A295424 _Marcin Barylski_, Feb 12 2018