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A253202 Number of essentially different ways of arranging the numbers 1 through 2n around a circle so that the sum of each pair of adjacent numbers is semiprime.

%I A253202 #10 Mar 27 2015 09:20:51
%S A253202 0,0,0,4,7,71,2555,24897,970556
%N A253202 Number of essentially different ways of arranging the numbers 1 through 2n around a circle so that the sum of each pair of adjacent numbers is semiprime.
%C A253202 Conjecture: a(n) > 0 for all n > 3.
%C A253202 This implies the semiprime conjecture, and it is similar to the prime circle problem mentioned in A051252.
%e A253202 Four arrangements for 2n = 8 are:
%e A253202 {1,3,6,8,7,2,4,5},
%e A253202 {1,3,7,2,8,6,4,5},
%e A253202 {1,5,4,2,7,3,6,8},
%e A253202 {1,5,4,6,3,7,2,8}, so a(4) = 4;
%e A253202 Seven arrangements for 2n = 10 are:
%e A253202 {1,3,6,8,7,2,4,10,5,9},
%e A253202 {1,3,6,9,5,10,4,2,7,8},
%e A253202 {1,3,7,2,4,10,5,9,6,8},
%e A253202 {1,3,7,2,8,6,4,10,5,9},
%e A253202 {1,5,10,4,2,8,7,3,6,9},
%e A253202 {1,8,2,7,3,6,4,10,5,9},
%e A253202 {1,8,6,3,7,2,4,10,5,9}, so a(5) = 7;
%t A253202 $RecursionLimit=500; try[lev_] := Module[{t, j}, If[lev>2n, (*then make sure the sum of the first and last is semiprime*) If[Plus@@Last/@FactorInteger [soln[[1]]+soln[[2n]]]==2&&soln[[2]]<=soln[[2n]], (*Print[soln]; *) cnt++ ], (*else append another number to the soln list*) t=soln[[lev-1]]; For[j=1, j<=Length[s[[t]]], j++, If[ !MemberQ[soln, s[[t]][[j]]], soln[[lev]]=s[[t]][[j]]; try[lev+1]; soln[[lev]]=0]]]]; For[lst={}; n=1, n<=7, n++, s=Table[{}, {2n}]; For[i=1, i<=2n, i++, For[j=1, j<=2n, j++, If[i!=j&& Plus@@Last/@FactorInteger [i+j]==2, AppendTo[s[[i]], j]]]]; soln=Table[0, {2n}]; soln[[1]]=1; cnt=0; try[2]; AppendTo[lst, cnt]]; lst
%Y A253202 Cf. A051252.
%K A253202 nonn,more
%O A253202 1,4
%A A253202 _Michel Lagneau_, Mar 25 2015