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%I #32 Apr 23 2018 11:46:36
%S 1,1,1,1,5,9,17,84,30,127,791,2404,11454,27680,25942,137272,515947,
%T 2834056,26583034,82099932,306004652,4518630225,11242369312,
%U 8942966426,95473633156,533328765065
%N Number of undirected circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that all the 2*n+2 numbers |i_0 +/- i_1|, |i_1 +/- i_2|, ..., |i_{n-1} +/- i_n|, |i_n +/- i_0| have the form (p-1)/2 with p an odd prime.
%C Conjecture: a(n) > 0 for all n > 0.
%C Note that if i-j = (p-1)/2 and i+j = (q-1)/2 for some odd primes p and q then 4*i+2 is the sum of the two primes p and q. So the conjecture is related to Goldbach's conjecture.
%C Zhi-Wei Sun also made the following similar conjecture: For any integer n > 5, there exists a circular permutation i_0, i_1, ..., i_n of 0, 1, ..., n such that all the 2*n+2 numbers 2*|i_k-i_{k+1}|+1 and 2*(i_k+i_{k+1})-1 (k = 0,...,n) (with i_{n+1} = i_0) are primes.
%H Z.-W. Sun, <a href="http://arxiv.org/abs/1309.1679">Some new problems in additive combinatorics</a>, arXiv preprint arXiv:1309.1679 [math.NT], 2013-2014.
%e a(n) = 1 for n = 1,2,3 due to the natural circular permutation (0,...,n).
%e a(4) = 1 due to the circular permutation (0,1,4,2,3).
%e a(5) = 5 due to the circular permutations (0,1,2,4,5,3), (0,1,4,2,3,5), (0,1,4,5,3,2), (0,2,1,4,5,3), (0,3,2,1,4,5).
%e a(6) = 9 due to the circular permutations
%e (0,1,2,4,5,3,6), (0,1,2,4,5,6,3), (0,1,4,2,3,5,6),
%e (0,1,4,2,3,6,5), (0,1,4,5,6,3,2), (0,2,1,4,5,3,6),
%e (0,2,1,4,5,6,3), (0,3,2,1,4,5,6), (0,5,4,1,2,3,6).
%e a(7) = 17 due to the circular permutations
%e (0,1,2,7,4,5,3,6), (0,1,2,7,4,5,6,3), (0,1,4,7,2,3,5,6),
%e (0,1,4,7,2,3,6,5), (0,1,7,2,4,5,3,6), (0,1,7,2,4,5,6,3),
%e (0,1,7,4,2,3,5,6), (0,1,7,4,2,3,6,5), (0,1,7,4,5,6,3,2),
%e (0,2,1,7,4,5,3,6), (0,2,1,7,4,5,6,3), (0,2,7,1,4,5,3,6),
%e (0,2,7,1,4,5,6,3), (0,3,2,1,7,4,5,6), (0,3,2,7,1,4,5,6),
%e (0,5,4,1,7,2,3,6), (0,5,4,7,1,2,3,6).
%t (* A program to compute required circular permutations for n = 7. To get "undirected" circular permutations, we should identify a circular permutation with the one of the opposite direction; for example, (0,6,3,5,4,7,2,1) is identical to (0,1,2,7,4,5,3,6) if we ignore direction. Thus a(7) is half of the number of circular permutations yielded by this program. *)
%t p[i_,j_]:=PrimeQ[2*Abs[i-j]+1]&&PrimeQ[2(i+j)+1]
%t V[i_]:=Part[Permutations[{1,2,3,4,5,6,7}],i]
%t m=0
%t Do[Do[If[p[If[j==0,0,Part[V[i],j]],If[j<7,Part[V[i],j+1],0]]==False,Goto[aa]],{j,0,7}]; m=m+1;Print[m,":"," ",0," ",Part[V[i],1]," ",Part[V[i],2]," ",Part[V[i],3]," ",Part[V[i],4]," ",Part[V[i],5]," ",Part[V[i],6]," ",Part[V[i],7]];Label[aa];Continue,{i,1,7!}]
%Y Cf. A000040, A051252, A002375, A228917, A228886.
%K nonn,more,hard
%O 1,5
%A _Zhi-Wei Sun_, Sep 09 2013
%E a(10)-a(26) from _Max Alekseyev_, Sep 17 2013
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