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A237638 a(n) is the number of prime sets such that each set contains enough prime numbers to decompose every even number from 6 to 2n into the sum of two of its elements (reuse allowed), while none of the sets is a subset of another such set.

%I A237638 #13 May 17 2014 03:27:45
%S A237638 1,1,1,1,1,2,2,3,4,4,5,6,6,9,11,11,11,13,16,23,25,31,47,57,63,70,74,
%T A237638 79,82,122,131,129,180,215,219,323,367,446,501,531,661,867,897,1311,
%U A237638 1471,1691,1695,2130,2288,2833,3363,3891,5435,8068,8867,13476,15451,15897
%N A237638 a(n) is the number of prime sets such that each set contains enough prime numbers to decompose every even number from 6 to 2n into the sum of two of its elements (reuse allowed), while none of the sets is a subset of another such set.
%H A237638 Lei Zhou, <a href="/A237638/b237638.txt">Table of n, a(n) for n = 3..79</a>
%e A237638 n=4, 2n=8. There is only one set of primes {3,5} such that 6=3+3, 8=3+5. So a(4)=1.
%e A237638 ...
%e A237638 n=8, 2n=16. We can find two sets, {3,5,7,11} and {3,5,7,13} that have such features. So a(8)=2. Here any set with more primes either contains an unused prime number or one of these two sets is a subset of them, like {3,5,7,11,13}, and thus is not considered. So a(8)=2.
%e A237638 ...
%e A237638 n=13, 2n=26. Five such sets are found: {3,5,7,11,13}, {3,5,7,13,17},{3,5,7,13,19}, {3,5,7,11,17,19}, {3,5,7,11,17,23}. So a(13)=5.
%t A237638 a = {{{3}}}; Table[n2 = 2*n; na = {}; la = Last[a]; lo = Length[la]; Do[ok = 0; Do[p1 = la[[i, j]]; p2 = n2 - p1; If[MemberQ[la[[i]], p2], ok = 1], {j, 1, Length[la[[i]]]}];
%t A237638   If[ok == 1, na = Sort[Append[na, la[[i]]]], Do[p1 = la[[i, j]]; p2 = n2 - p1; If[PrimeQ[p2], ng = Sort[Append[la[[i]], p2]]; big = 0; If[Length[na] > 0, Do[If[Intersection[na[[k]], ng] == na[[k]], big = 1], {k, 1, Length[na]}]]; If[big == 0, na = Sort[Append[na, ng]]]], {j, 1, Length[la[[i]]]}]], {i, 1, lo}]; AppendTo[a, na]; Length[na], {n, 4, 60}](* Program lists the 4th item and beyond *)
%Y A237638 Cf. A000040, A002375, A240708, A237628.
%K A237638 nonn,hard
%O A237638 3,6
%A A237638 _Lei Zhou_, May 02 2014