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A229232 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. 0
0, 0, 0, 1, 0, 2, 1, 2, 2, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
COMMENTS
Conjecture: a(n) > 0 for all n > 5 with n not equal to 13.
Zhi-Wei Sun also made the following conjectures:
(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.
(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.
LINKS
EXAMPLE
a(4) = 1 due to the circular permutation (1,3,2,4).
a(6) = 2 due to the circular permutations
(1,3,2,4,5,6) and (1,3,2,6,5,4).
a(7) = 1 due to the circular permutation (1,3,2,7,6,5,4).
a(8) = 2 due to the circular permutations
(1,3,2,7,6,5,4,8) and (1,4,5,6,7,2,3,8).
a(9) = 2 due to the circular permutations
(1,3,4,5,6,7,2,9,8) and (1,3,8,9,2,7,6,5,4).
a(10) = 8 due to the circular permutations
(1,3,4,5,6,7,2,9,10,8), (1,3,4,5,6,7,2,10,9,8),
(1,3,8,9,10,2,7,6,5,4), (1,3,8,10,9,2,7,6,5,4),
(1,3,10,8,9,2,7,6,5,4), (1,3,10,9,2,7,6,5,4,8),
(1,4,5,6,7,2,3,10,9,8), (1,4,5,6,7,2,9,10,3,8).
a(13) = 0 since 8 is the unique j among 1 ,..., 12 with 13*j-1 prime.
MATHEMATICA
(* 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. *)
V[i_]:=V[i]=Part[Permutations[{2, 3, 4, 5, 6, 7, 8}], i]
f[i_, j_]:=f[i, j]=PrimeQ[i*j-1]
m=0
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}];
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!}]
CROSSREFS
Sequence in context: A145863 A240492 A110775 * A092186 A138262 A276990
KEYWORD
nonn,more,hard
AUTHOR
Zhi-Wei Sun, Sep 16 2013
STATUS
approved

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Last modified July 18 21:02 EDT 2024. Contains 374388 sequences. (Running on oeis4.)