

A322921


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


0



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
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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



FORMULA



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



STATUS

approved



