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 A229005 Number of undirected circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that all the n+1 numbers |i_0^2-i_1^2|, |i_1^2-i_2^2|, ..., |i_{n-1}^2-i_n^2|, |i_n^2-i_0^2| are of the form (p-1)/2 with p an odd prime. 5
 1, 0, 1, 0, 1, 6, 3, 16, 18, 122, 97, 2725, 26457, 10615, 367132, 158738, 1356272, 72423339 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Conjecture: a(n) > 0 except for n = 2, 4. LINKS Zhi-Wei Sun, Some new problems in additive combinatorics, preprint, arXiv:1309.1679 [math.NT], 2013-2014. EXAMPLE a(1) = 1 due to the circular permutation (0,1). a(2) = 0 since 2*2^2+1 is composite. a(3) = 1 due to the circular permutation (0,1,2,3). a(4) = 0 since 2*(4^2-k^2)+1 is composite for any k = 0,2,3. a(5) = 1 due to the circular permutation (0,1,4,5,2,3). a(6) = 6 due to the circular permutations (0,1,3,2,5,4,6), (0,1,4,6,5,2,3), (0,1,6,4,5,2,3), (0,3,1,2,5,4,6), (0,3,2,1,4,5,6), (0,3,2,5,4,1,6). a(7) = 3 due to the circular permutations (0,1,7,4,6,5,2,3), (0,3,2,1,7,4,5,6), (0,3,2,5,4,7,1,6). a(8) = 16 due to the circular permutations (0,1,3,2,5,8,7,4,6), (0,1,6,4,7,8,5,2,3), (0,1,7,8,4,6,5,2,3), (0,1,8,7,4,6,5,2,4), (0,3,1,2,5,8,7,4,6), (0,3,2,1,4,7,8,5,6), (0,3,2,1,7,4,8,5,6), (0,3,2,1,7,8,4,5,6), (0,3,2,1,7,8,5,4,6), (0,3,2,1,8,7,4,5,6), (0,3,2,5,4,7,8,1,6), (0,3,2,5,4,8,7,1,6), (0,3,2,5,8,1,7,4,6), (0,3,2,5,8,4,7,1,6), (0,3,2,5,8,7,1,4,6), (0,3,2,5,8,7,4,1,6). a(9) > 0 due to the permutation (0,3,2,1,6,4,7,8,5,9). a(10) > 0 due to the permutation (0,9,5,6,4,7,8,10,2,3,1). 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, 1, 7, 4, 5, 2, 3) is identical to (0, 3, 2, 5, 4, 7, 1, 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^2-j^2]+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 Cf. A000040, A051252, A228917, A228956, A228886. Sequence in context: A267831 A328011 A231881 * A067990 A174012 A050008 Adjacent sequences: A229002 A229003 A229004 * A229006 A229007 A229008 KEYWORD nonn,more,hard AUTHOR Zhi-Wei Sun, Sep 10 2013 EXTENSIONS a(10)-a(18) from Alois P. Heinz, Sep 10 2013 STATUS approved

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Last modified January 28 12:12 EST 2023. Contains 359869 sequences. (Running on oeis4.)