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A229130 Number of permutations i_0, i_1, ..., i_n of 0, 1, ..., n with i_0 = 0 and i_n = n such that the n+1 numbers i_0^2+i_1, i_1^2+i_2, ..., i_{n-1}^2+i_n, i_n^2+i_0 are all relatively prime to both n-1 and n+1. 3
1, 0, 1, 1, 0, 6, 3, 42, 68, 2794, 0, 5311604, 478, 57009, 2716452, 10778632, 207360 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

Conjecture: a(n) > 0 except for n = 2, 5, 11. Similarly, for any positive integer n not equal to 4, there is a permutation i_0, i_1, ..., i_n of 0, 1, ..., n with i_0 = 0 and i_n = n such that the n+1 numbers i_0^2-i_1, i_1^2-i_2, ..., i_{n-1}^2-i_n, i_n^2-i_0 are all coprime to both n-1 and n+1.

Zhi-Wei Sun also made the following general conjecture:

  For any positive integer k, define E(k) to be the set of those positive integers n for which there is no permutation i_0, i_1, ..., i_n of 0, 1, ..., n with i_0 = 0 and i_n = n such that all the n+1 numbers i_0^k+i_1, i_1^k+i_2, ..., i_{n-1}^k+i_n, i_n^k+i_0 are coprime to both n-1 and n+1. Then E(k) is always finite; in particular, E(1) = {2,4}, E(2) = {2,5,11} and E(3) = {2,4}.

a(19) = 106144, a(23) = 559350576. - Alois P. Heinz, Sep 16 2013

LINKS

Table of n, a(n) for n=1..17.

Zhi-Wei Sun, List of required permutations for n = 1..10

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

EXAMPLE

a(3) = 1 due to the permutation (i_0,i_1,i_2,i_3)=(0,1,2,3).

a(4) = 1 due to the permutation (0,1,3,2,4).

a(6) = 1 due to the permutations

  (0,1,3,2,5,4,6), (0,1,3,4,2,5,6), (0,2,5,1,3,4,6),

  (0,3,2,4,1,5,6), (0,3,4,1,2,5,6), (0,4,1,3,2,5,6).

a(7) = 3 due to the permutations

  (0,1,6,5,4,3,2,7), (0,5,4,3,2,1,6,7), (0,5,6,1,4,3,2,7).

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

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

a(10) > 0 due to the permutation (0,1,3,5,4,7,9,8,6,2,10).

a(11) = 0 since 6 is the unique i among 0,...,11 with i^2+5 coprime to 11^2-1, and it is also the unique j among 1,...,10 with j^2+11 coprime to 11^2-1.

MATHEMATICA

(* A program to compute required permutations for n = 8. *)

V[i_]:=Part[Permutations[{1, 2, 3, 4, 5, 6, 7}], i]

m=0

Do[Do[If[GCD[If[j==0, 0, Part[V[i], j]]^2+If[j<7, Part[V[i], j+1], 8], 8^2-1]>1, 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], " ", 8]; Label[aa]; Continue, {i, 1, 7!}]

CROSSREFS

Cf. A228886, A229082, A229038, A229005, A228917, A228956.

Sequence in context: A202363 A038257 A090138 * A088390 A286782 A054380

Adjacent sequences:  A229127 A229128 A229129 * A229131 A229132 A229133

KEYWORD

nonn,more,hard

AUTHOR

Zhi-Wei Sun, Sep 15 2013

EXTENSIONS

a(12)-a(17) from Alois P. Heinz, Sep 15 2013

STATUS

approved

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Last modified October 14 18:28 EDT 2019. Contains 328022 sequences. (Running on oeis4.)