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

1,3

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.

Links

Table of n, a(n) for n=1..70.

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

Cf. A002375, A045917.

Sequence in context: A047740 A137687 A024745 * A030581 A113609 A206916

Adjacent sequences: A322918 A322919 A322920 * A322922 A322923 A322924

Keyword

nonn

Author

Pedro Caceres, Dec 30 2018

Status

approved