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%I #11 Apr 24 2014 09:33:10
%S 0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,0,3,4,2,2,4,2,3,5,3,3,5,2,4,6,2,4,5,0,
%T 4,5,4,4,7,3,4,6,3,4,9,3,5,7,3,5,7,3,5,7,5,6,10,5,6,12,2,5,10,2,6,7,4,
%U 4,4,5,7,9,6,5,11,0,6,10,3,7,8,4,4,13,8
%N Number of decompositions of 2n=p1+p2 (prime p1 <= p2), where at least one other such pair 2n=p3+p4 (prime p3 <= p4) exists such that |p1-p3|= 6 or 12.
%C It is conjectured that a(n)=0 only when n=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16, 34, 76, 229.
%C The 2n decompositions counted in this sequence are a subset of 2n decompositions as of in Goldbach conjecture (A002375).
%C Per definition, all nonzero terms are greater than 1.
%H Lei Zhou, <a href="/A240713/b240713.txt">Table of n, a(n) for n = 1..10000</a>
%e For n = 11, 2n=22. 22 = 3 + 19 = 5 + 17 = 11 + 11. |5-11|=6 so pair 5+17 and 11+11 are counted. So a(11)=2.
%e ...
%e For n = 17, 2n=34. 34 = 3 + 31 = 5 + 29 = 11 + 23 = 17 + 17. |5-11|=6, so 5+29 and 11+23 are counted. Also since |11-17|=6, 17+17 is also counted (where 11+23 is already counted). In case |5-17|=12, both instances are already counted. So overall three instances are found. a(17)=3.
%t Table[s = 2*n; ct = 0; p = 1; While[p = NextPrime[p]; p <= n, If[PrimeQ[s - p], ok = 0; a1 = p - 12; b1 = s - a1; a2 = p - 6; b2 = s - a2; a3 = p + 6; b3 = s - a3; a4 = p + 12; b4 = s - a4; If[a1 > 0, If[PrimeQ[a1] && PrimeQ[b1], ok = 1]]; If[a2 > 0, If[PrimeQ[a2] && PrimeQ[b2], ok = 1]]; If[a3 <= n, If[PrimeQ[a3] && PrimeQ[b3], ok = 1]]; If[a4 <= n, If[PrimeQ[a4] && PrimeQ[b4], ok = 1]]; If[ok == 1, ct++]]]; ct, {n, 1, 85}]
%Y Cf. A000040, A002375, A240712.
%K nonn,easy
%O 1,11
%A _Lei Zhou_, Apr 10 2014
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