A294185 -id:A294185 - cp's OEIS frontend

A294186 Number of distinct greater twin primes which are in Goldbach partitions of 2n.

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

Offset: 1

Marcin Barylski, Feb 11 2018

nonn

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.
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).
Is a(n) = A294185(n-1)? - R. J. Mathar, Mar 22 2024

			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).

Marcin Barylski, Plot of first 20000 elements of the A294186
Marcin Barylski, C++ program for generating A294186
Tomas Oliveira e Silva, Goldbach conjecture verification

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

isgtwin(p) = isprime(p) && isprime(p-2);
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

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

Original entry on oeis.org

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

Marcin Barylski, Feb 12 2018

nonn

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).

			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.

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

Cf. A294186, A294185, A001097.

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

cp's OEIS Frontend

A294186 Number of distinct greater twin primes which are in Goldbach partitions of 2n.

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