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%I #18 Feb 11 2019 19:40:07
%S 1,1,2,3,3,4,4,5,5,6,6,6,7,8,9,7,8,8,10,12,10,9,8,11,12,11,10,13,11,
%T 14,13,11,13,14,19,13,11,12,15,18,16,16,14,16,19,16,16,17,19,21,15,17,
%U 15,20,24,19,17,16,20,22,18,18,22,19,27,21,17,20,21,30
%N From Goldbach's conjecture: a(n) is the number of decompositions of 6n into a sum of two primes.
%C According to Goldbach's conjecture all even numbers can be decomposed into one or more sums of two prime numbers.
%C Each even number N belongs to one of the following sets: {N == 0 (mod 6)}, {(N + 2) == 0 (mod 6)}, and {(N - 2) == 0 (mod 6)}.
%C Conjecture: In any combination of three consecutive even numbers >= 48, the one of the form N == 0 (mod 6) will have the largest number of decompositions into 2 prime numbers. This sequence contains those local maxima for every set of three consecutive even numbers. This sequence forms the upper envelope of Goldbach's comet chart.
%F a(n) = A002375(3*n).
%e a(1) = 1 because 6 * 1 = 6 can be decomposed as (3 + 3);
%e a(8) = 5 is the number of ways that 6 * 8 = 48 can be decomposed into sums of two prime numbers: 5 + 43, 11 + 37, 17 + 31, 29 + 19, 41 + 7.
%t Table[Count[IntegerPartitions[6n, {2}], _?(AllTrue[#, PrimeQ] && FreeQ[#, 2]&)], {n, 100}] (* _Alonso del Arte_, Dec 31 2018, just a tiny modification of _Harvey P. Dale_'s for A002375 *)
%Y Cf. A002375, A045917.
%K nonn
%O 1,3
%A _Pedro Caceres_, Dec 30 2018