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Numbers n such that the number of decompositions of 2n into sum of two primes (counting 1 as a prime) is 1 or a composite.
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%I #12 Feb 28 2020 03:32:35

%S 1,12,15,17,18,22,23,24,25,27,29,31,33,37,42,44,45,46,49,50,51,52,53,

%T 54,58,59,60,61,63,64,66,67,69,70,71,73,77,79,80,81,82,83,84,85,86,87,

%U 90,92,95,96,97,98,99,100,101,102,107,110,112,115,117,118,119

%N Numbers n such that the number of decompositions of 2n into sum of two primes (counting 1 as a prime) is 1 or a composite.

%C Arises in Goldbach conjecture.

%F {integers n > 0 such that A001031(n) is in A018252} = {integers n > 0 such that A001031(n) is not in A000040}.

%e 1 is a term because there is a unique decomposition of 2*1 = 2 into a sum of two primes (counting 1 as a prime), namely 2 = 1 + 1.

%e 12 is a term because there are 4 decompositions of 2*12 = 24 into a sum of two primes (counting 1 as a prime), namely 1 + 23, 5 + 19, 7 + 17, and 11 + 13, and 4 is a composite number.

%o (Sage)

%o def is_A188766(n):

%o pp = set(prime_range(2*n+1)+[1])

%o return not is_prime(len([x for x in Partitions(2*n,length=2) if set(x) <= pp]))

%o # _D. S. McNeil_, Apr 10 2011

%Y Cf. A000040, A001031, A002808, A018252.

%K nonn,easy

%O 1,2

%A _Jonathan Vos Post_, Apr 09 2011