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%I #19 Aug 04 2019 11:40:58
%S 0,0,0,1,0,2,1,2,2,8
%N Number of undirected circular permutations pi(1), ..., pi(n) of 1, ..., n with the n numbers pi(1)*pi(2)-1, pi(2)*pi(3)-1, ..., pi(n-1)*pi(n)-1, pi(n)*pi(1)-1 all prime.
%C Conjecture: a(n) > 0 for all n > 5 with n not equal to 13.
%C Zhi-Wei Sun also made the following conjectures:
%C (1) For any integer n > 1, there is a permutation pi(1), ..., pi(n) of 1, ..., n such that the n numbers 2*pi(1)*pi(2)-1, ..., 2*pi(n-1)*pi(n)-1, 2*pi(n)*pi(1)-1 are all prime. Also, for any positive integer n not equal to 4, there is a permutation pi(1), ..., pi(n) of 1, ..., n such that the n numbers 2*pi(1)*pi(2)+1, ..., 2*pi(n-1)*pi(n)+1, 2*pi(n)*pi(1)+1 are all prime.
%C (2) Let F be a finite field with q > 7 elements. Then, there is a circular permutation a_1,...,a_{q-1} of the q-1 nonzero elements of F such that all the q-1 elements a_1*a_2-1, a_2*a_3-1, ..., a_{q-2}*a_{q-1}-1, a_{q-1}*a_1-1 are primitive elements of the field F (i.e., generators of the multiplicative group F\{0}). Also, there is a circular permutation b_1,...,b_{q-1} of the q-1 nonzero elements of F such that all the q-1 elements b_1*b_2+1, b_2*b_3+1, ..., b_{q-2}*b_{q-1}+1, b_{q-1}*b_1+1 are primitive elements of the field F.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1309.1679">Some new problems in additive combinatorics, preprint, arXiv:1309.1679 [math.NT], 2013-2014.
%e a(4) = 1 due to the circular permutation (1,3,2,4).
%e a(6) = 2 due to the circular permutations
%e (1,3,2,4,5,6) and (1,3,2,6,5,4).
%e a(7) = 1 due to the circular permutation (1,3,2,7,6,5,4).
%e a(8) = 2 due to the circular permutations
%e (1,3,2,7,6,5,4,8) and (1,4,5,6,7,2,3,8).
%e a(9) = 2 due to the circular permutations
%e (1,3,4,5,6,7,2,9,8) and (1,3,8,9,2,7,6,5,4).
%e a(10) = 8 due to the circular permutations
%e (1,3,4,5,6,7,2,9,10,8), (1,3,4,5,6,7,2,10,9,8),
%e (1,3,8,9,10,2,7,6,5,4), (1,3,8,10,9,2,7,6,5,4),
%e (1,3,10,8,9,2,7,6,5,4), (1,3,10,9,2,7,6,5,4,8),
%e (1,4,5,6,7,2,3,10,9,8), (1,4,5,6,7,2,9,10,3,8).
%e a(13) = 0 since 8 is the unique j among 1 ,..., 12 with 13*j-1 prime.
%t (* A program to compute required circular permutations for n = 8. To get "undirected" circular permutations, we should identify a circular permutation with the one of the opposite direction; for example, (1,8,4,5,6,7,2,3) is identical to (1,3,2,7,6,5,4,8) if we ignore direction. Thus, a(8) is half of the number of circular permutations yielded by this program. *)
%t V[i_]:=V[i]=Part[Permutations[{2,3,4,5,6,7,8}],i]
%t f[i_,j_]:=f[i,j]=PrimeQ[i*j-1]
%t m=0
%t Do[Do[If[f[If[j==0,1,Part[V[i],j]],If[j<7,Part[V[i],j+1],1]]==False,Goto[aa]],{j,0,7}];
%t m=m+1;Print[m,":"," ",1," ",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. A051252, A227456, A228917, A228956, A229082.
%K nonn,more,hard
%O 1,6
%A _Zhi-Wei Sun_, Sep 16 2013
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