A229141 Number of circular permutations i_1, ..., i_n of 1, ..., n such that all the n sums i_1^2+i_2, ..., i_{n-1}^2+i_n, i_n^2+i_1 are among those integers m with the Jacobi symbol (m/(2n+1)) equal to 1.

1, 0, 0, 2, 0, 1, 0, 5, 35, 0

Offset

1,4

Comments

Conjecture: a(n) > 0 if 2*n+1 is a prime greater than 11.

Zhi-Wei Sun also made the following conjectures:

(1) For any prime p = 2*n+1 > 11, there is a circular permutation i_1, ..., i_n of 1, ..., n such that all the n numbers i_1^2-i_2, i_2^2-i_3, ..., i_{n-1}^2-i_n, i_n^2-i_1 are quadratic residues modulo p.

(2) Let p = 2*n+1 be an odd prime. If p > 13 (resp., p > 11), then there is a circular permutation i_1, ..., i_n of 1, ..., n such that all the n numbers i_1^2+i_2, i_2^2+i_3, ..., i_{n-1}^2+i_n, i_n^2+i_1 (resp., the n numbers i_1^2-i_2, i_2^2-i_3, ..., i_{n-1}^2-i_n, i_n^2-i_1) are primitive roots modulo p.

(3) Let p = 2*n+1 be an odd prime. If p > 19 (resp. p > 13), then there is a circular permutation i_1, ..., i_n of 1, ..., n such that all the n numbers i_1^2+i_2^2, i_2^2+i_3^2, ..., i_{n-1}^2+i_n^2, i_n^2+i_1^2 (resp., the n numbers i_1^2-i_2^2, i_2^2-i_3^2, ..., i_{n-1}^2-i_n^2, i_n^2-i_1^2) are primitive roots modulo p.

See also the linked arXiv paper of Sun for more conjectures involving primitive roots modulo primes.

Links

Table of n, a(n) for n=1..10.

Zhi-Wei Sun, Some new problems in additive combinatorics, preprint, arXiv:1309.1679 [math.NT], 2013-2014.

Example

a(4) = 2 due to the permutations (1,3,2,4) and (1,4,3,2).
a(6) = 1 due to the permutation (1,3,5,2,6,4).
a(8) = 5 due to the permutations
   (1,3,4,2,5,8,6,7), (1,8,3,6,2,4,5,7), (1,8,3,6,7,4,2,5),
   (1,8,3,7,6,2,4,5), (1,8,6,7,3,4,2,5).
a(9) > 0 due to the permutation (1,3,7,6,8,4,9,2,5).

Mathematica

(* A program to compute the desired circular permutations for n = 8. *)
f[i_, j_, p_]:=f[i, j, p]=JacobiSymbol[i^2+j, p]==1
V[i_]:=Part[Permutations[{2, 3, 4, 5, 6, 7, 8}], i]
m=0
Do[Do[If[f[If[j==0, 1, Part[V[i], j]], If[j<7, Part[V[i], j+1], 1], 17]==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

Cf. A229038, A229082, A229130, A229005.

Sequence in context: A196777 A318361 A078924 * A137526 A137525 A166335

Adjacent sequences: A229138 A229139 A229140 * A229142 A229143 A229144

Keyword

nonn,more,hard

Author

Zhi-Wei Sun, Sep 15 2013

Extensions

a(10) = 0 from R. J. Mathar, Sep 15 2013

Status

approved