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A294186
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Number of distinct greater twin primes which are in Goldbach partitions of 2n.
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3
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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
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OFFSET
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1,5
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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).
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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