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

1,4

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).

Links

Table of n, a(n) for n=1..83.

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

Example

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.

Prog

(C++) // See Barylski link.
(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

Crossrefs

Cf. A294186, A294185, A001097.

Sequence in context: A081388 A231070 A230252 * A002372 A035026 A224962

Adjacent sequences: A295421 A295422 A295423 * A295425 A295426 A295427

Keyword

nonn

Author

Marcin Barylski, Feb 12 2018

Status

approved