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A276034 a(n) is the number of decompositions of 2n into an unordered sum of two primes in A274987.

%I A276034 #15 Nov 21 2018 02:34:18
%S A276034 0,0,1,1,2,1,2,2,2,2,2,2,2,2,2,0,3,2,1,2,2,2,1,2,1,0,2,1,1,2,2,3,3,2,
%T A276034 2,2,2,3,2,1,2,4,3,1,5,3,2,5,1,2,2,2,5,2,3,4,5,3,2,5,2,1,4,0,1,5,3,1,
%U A276034 3,5,4,4,3,2,4,3,3,4,2,3,7,2,2,3,2,2,2
%N A276034 a(n) is the number of decompositions of 2n into an unordered sum of two primes in A274987.
%C A276034 The two primes are allowed to be the same.
%C A276034 It is conjectured that the primes in A274987 (a subset of all primes) are sufficient to decomposite even numbers into two primes in A274987 when n > 958.
%C A276034 This sequence provides a very tight alternative of the Goldbach conjecture for all positive integers, in which indices of zero terms form a complete sequence {1, 2, 16, 26, 64, 97, 107, 122, 146, 167, 194, 391, 451, 496, 707, 856, 958}.
%C A276034 There is no more zero terms of a(n) tested up to n = 100000.
%H A276034 Lei Zhou, <a href="/A276034/b276034.txt">Table of n, a(n) for n = 1..10000</a>
%H A276034 Lei Zhou, <a href="/A276034/a276034.png">List plot of the first 10000 terms of a(n)</a>.
%e A276034 A274987 = {3, 5, 7, 11, 13, 17, 23, 31, 37, 53, 59, 61, 73, 79, 83, 89, 101, 103, 109, ...}.
%e A276034 For n=3, 2n=6 = 3+3, one case of decomposition, so a(3)=1;
%e A276034 for n=4, 2n=8 = 3+5, one case of decomposition, so a(4)=1;
%e A276034 ...
%e A276034 for n=17, 2n=34 = 3+31 = 11+23 = 17+17, three cases of decompositions, so a(17)=3.
%t A276034 p = 3; sp = {p}; a = Table[m = 2*n; l = Length[sp]; While[sp[[l]] < m, While[p = NextPrime[p]; cp = 2*3^(Floor[Log[3, 2*p - 1]]) - p; ! PrimeQ[cp]]; AppendTo[sp, p]; l++]; ct = 0; Do[If[(2*sp[[i]] <= m) && (MemberQ[sp, m - sp[[i]]]), ct++], {i, 1, l}]; ct, {n, 1, 87}]
%Y A276034 Cf. A002375, A045917, A001031, A274987, A171611, A240708, A240712, A230443.
%K A276034 nonn,easy,base,look
%O A276034 1,5
%A A276034 _Lei Zhou_, Nov 15 2016