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 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified September 30 17:50 EDT 2023. Contains 365792 sequences. (Running on oeis4.)