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A228956
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.
6
1, 1, 1, 1, 5, 9, 17, 84, 30, 127, 791, 2404, 11454, 27680, 25942, 137272, 515947, 2834056, 26583034, 82099932, 306004652, 4518630225, 11242369312, 8942966426, 95473633156, 533328765065
OFFSET
1,5
COMMENTS
Conjecture: a(n) > 0 for all n > 0.
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.
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.
LINKS
Z.-W. Sun, Some new problems in additive combinatorics, arXiv preprint arXiv:1309.1679 [math.NT], 2013-2014.
EXAMPLE
a(n) = 1 for n = 1,2,3 due to the natural circular permutation (0,...,n).
a(4) = 1 due to the circular permutation (0,1,4,2,3).
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).
a(6) = 9 due to the circular permutations
(0,1,2,4,5,3,6), (0,1,2,4,5,6,3), (0,1,4,2,3,5,6),
(0,1,4,2,3,6,5), (0,1,4,5,6,3,2), (0,2,1,4,5,3,6),
(0,2,1,4,5,6,3), (0,3,2,1,4,5,6), (0,5,4,1,2,3,6).
a(7) = 17 due to the circular permutations
(0,1,2,7,4,5,3,6), (0,1,2,7,4,5,6,3), (0,1,4,7,2,3,5,6),
(0,1,4,7,2,3,6,5), (0,1,7,2,4,5,3,6), (0,1,7,2,4,5,6,3),
(0,1,7,4,2,3,5,6), (0,1,7,4,2,3,6,5), (0,1,7,4,5,6,3,2),
(0,2,1,7,4,5,3,6), (0,2,1,7,4,5,6,3), (0,2,7,1,4,5,3,6),
(0,2,7,1,4,5,6,3), (0,3,2,1,7,4,5,6), (0,3,2,7,1,4,5,6),
(0,5,4,1,7,2,3,6), (0,5,4,7,1,2,3,6).
MATHEMATICA
(* 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. *)
p[i_, j_]:=PrimeQ[2*Abs[i-j]+1]&&PrimeQ[2(i+j)+1]
V[i_]:=Part[Permutations[{1, 2, 3, 4, 5, 6, 7}], i]
m=0
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!}]
CROSSREFS
KEYWORD
nonn,more,hard
AUTHOR
Zhi-Wei Sun, Sep 09 2013
EXTENSIONS
a(10)-a(26) from Max Alekseyev, Sep 17 2013
STATUS
approved