OFFSET
1,3
COMMENTS
According to Goldbach's conjecture all even numbers can be decomposed into one or more sums of two prime numbers.
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)}.
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.
FORMULA
a(n) = A002375(3*n).
EXAMPLE
a(1) = 1 because 6 * 1 = 6 can be decomposed as (3 + 3);
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.
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Pedro Caceres, Dec 30 2018
STATUS
approved