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

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, 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, 4, 8, 12, 6, 3, 10, 9, 10, 15, 7, 7, 12, 7, 10, 14, 6, 9, 13, 5, 7

Offset: 1

Author

Marcin Barylski, Feb 12 2018

Keywords

• nonn

Comments

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.

Conjecture. Further empirical examinations lead to a hypothesis that all even numbers n > 4 have at least 1 twin prime in GP(n).

a(n) <= A294185(n) + A294186(n).

Examples

			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.
		

Links

• Marcin Barylski, Plot of first 20000 elements of the A295424
• Marcin Barylski, C++ program for generating A295424
• Marcin Barylski, Studies on Twin Primes in Goldbach Partitions of Even Numbers
• Tomas Oliveira e Silva, Goldbach conjecture verification

Crossrefs

Cf. A294186, A294185, A001097.

Programs

PARI

istwin(p) = isprime(p) && (isprime(p-2) || isprime(p+2));
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