OFFSET
1,5
COMMENTS
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
LINKS
Marcin Barylski, Plot of first 20000 elements of the A294186
Marcin Barylski, C++ program for generating A294186
Tomas Oliveira e Silva, Goldbach conjecture verification
EXAMPLE
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).
PROG
(C++) See Barylski link.
(PARI) 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
CROSSREFS
KEYWORD
nonn
AUTHOR
Marcin Barylski, Feb 11 2018
STATUS
approved