A322921 From Goldbach's conjecture: a(n) is the number of decompositions of 6n into a sum of two primes.

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, 14, 13, 11, 13, 14, 19, 13, 11, 12, 15, 18, 16, 16, 14, 16, 19, 16, 16, 17, 19, 21, 15, 17, 15, 20, 24, 19, 17, 16, 20, 22, 18, 18, 22, 19, 27, 21, 17, 20, 21, 30

Offset: 1

Author

Pedro Caceres, Dec 30 2018

Keywords

• nonn

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.

Examples

			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.
		

Crossrefs

Cf. A002375, A045917.

Programs

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 *)

Formula

a(n) = A002375(3*n).