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.

1, 0, 1, 0, 1, 6, 3, 16, 18, 122, 97, 2725, 26457, 10615, 367132, 158738, 1356272, 72423339

Offset

1,6

Comments

Conjecture: a(n) > 0 except for n = 2, 4.

Links

Table of n, a(n) for n=1..18.

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