

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_{n1}^2+i_n, i_n^2+i_0 are all relatively prime to both n1 and n+1.


3



1, 0, 1, 1, 0, 6, 3, 42, 68, 2794, 0, 5311604, 478, 57009, 2716452, 10778632, 207360, 39187872956340, 106144, 26869397610, 11775466120, 22062519153360, 559350576, 29991180449906858400, 257272815600, 12675330087321600, 52248156883498208
(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_0^2i_1, i_1^2i_2, ..., i_{n1}^2i_n, i_n^2i_0 are all coprime to both n1 and n+1.
ZhiWei 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_{n1}^k+i_n, i_n^k+i_0 are coprime to both n1 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}.


LINKS

Table of n, a(n) for n=1..27.
ZhiWei Sun, List of required permutations for n = 1..10
ZhiWei Sun, Some new problems in additive combinatorics, preprint, arXiv:1309.1679 [math.NT], 20132014.


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 11^21, and it is also the unique j among 1,...,10 with j^2+11 coprime to 11^21.


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], 8^21]>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 * A349492 A088390 A286782
Adjacent sequences: A229127 A229128 A229129 * A229131 A229132 A229133


KEYWORD

nonn,hard


AUTHOR

ZhiWei Sun, Sep 15 2013


EXTENSIONS

a(12)a(17) from Alois P. Heinz, Sep 15 2013
a(19) and a(23) from Alois P. Heinz, Sep 16 2013
a(18), a(20)a(22) and a(24)a(27) from Bert Dobbelaere, Feb 18 2020


STATUS

approved



