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%I #8 May 01 2014 05:16:56
%S 0,0,1,1,2,1,2,2,2,2,3,3,3,2,3,2,4,4,2,3,4,3,4,5,4,3,5,3,4,6,3,5,6,2,
%T 5,6,5,5,7,4,5,8,5,4,9,4,5,7,3,6,8,5,6,8,6,7,10,6,6,12,4,5,10,3,7,9,6,
%U 5,8,7,8,11,6,5,12,4,8,11,5,8,10,5,6,13,9,6,11,7,7,14,6,8,13,5,8,11,7,9,13,8,9,14,7,7,19,6,7,12,6,9
%N Number of decompositions of 2n into an unordered sum of two terms of A240699.
%C The first different term of this sequence to A002375 is a(107).
%C Conjecture: for n >= 3, this sequence is always positive.
%C This is a stronger version of the Goldbach Conjecture.
%H Lei Zhou, <a href="/A240708/b240708.txt">Table of n, a(n) for n = 1..10000</a>
%e For n <= 106, refer to examples in A002375.
%e For n = 107, 2n=214. A240699 up to 214 gives {3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199}. We have 214 = 17+197 = 23+191 = 41+173 = 47+167 = 83+131 = 101+113 = 107+107. Seven instances found. So a(107)=7.
%e Where as for A002375, there is one more instance as 3+211, however 211 is not a term in A240699.
%t a240699 = {3}; Table[s = 2*n; While[a240699[[-1]] < s, p = a240699[[-1]]; While[p = NextPrime[p]; ((NextPrime[p] - p) > 6) && (6 < (p - NextPrime[p, -1]))]; AppendTo[a240699, p]]; pos = 0; ct = 0; While[pos++; pos <= Length[a240699], p = a240699[[pos]]; If[p <= n, If[MemberQ[a240699, s - p], ct++]]]; ct, {n, 1, 110}]
%Y Cf. A000040, A002375, A045917, A240699.
%K nonn,easy
%O 1,5
%A _Lei Zhou_, Apr 10 2014
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