A228956 -id:A228956 - cp's OEIS frontend

A228886 Number of permutations i_0,i_1,...,i_n of 0,1,...,n with i_0 = 0 and i_n = n, and with i_0+i_1, i_1+i_2, ..., i_{n-1}+i_n, i_n+i_0 all coprime to both n-1 and n+1.

1, 0, 1, 0, 1, 8, 3, 24, 78, 1164, 34, 4021156, 400, 87180, 2499480, 7509358, 1700352, 39982182134232, 1427688, 212987263960, 9533487948, 36638961135462, 29317847040, 30258969747586970112, 1655088666624

Offset: 1

Zhi-Wei Sun, Sep 07 2013

Conjecture: a(n) > 0 except for n = 2, 4.
Note that this conjecture is stronger than the one stated in the comments in A228860. Also, there is no permutation i_0, ..., i_7 of 0, ..., 7 with i_0+i_1, i_1+i_2, ..., i_6+i_7, i_7+i_0 all coprime to 7*13 - 1 = 90.
For n > 1 let S(n) be an undirected simple graph with vertices 0, 1, ..., n which has an edge connecting two distinct vertices i and j if and only if i + j is relatively prime to both n-1 and n+1. Then a(n) is just the number of those Hamiltonian cycles in S(n) on which the vertices 0 and n are adjacent.
We have some other similar conjectures. For example, for any positive integer n there is a permutation i_0,i_1,...,i_n of 0,1,...,n with i_0 = 0 and i_n = n such that i_0 + i_1, i_1 + i_2, ..., i_{n-1} + i_n, i_n + i_0 are all relatively prime to both 2*n-1 and 2*n+1.
On Sep 10 2013, Zhi-Wei Sun proved the conjecture for any positive odd integer n. In fact, if n == 1 or 3 (mod 6) then we may take the permutation 0,n-2,2,n-4,4,...,1,n-1,n which meets the requirement; if n == 3 or 5 (mod 6) then the permutation 0,1,n-1,3,n-3,...,n-2,2,n suffices for our purpose.

			a(2) = 0 since 1+2 = 2^2-1.
a(3) = 1 due to the identical permutation (0,1,2,3).
a(4) = 0 since 1+2 divides 4^2-1.
a(5) = 1 due to the permutation (0,1,4,3,2,5).
a(6) = 8 due to the permutations
  (0,1,2,4,5,3,6), (0,1,3,5,4,2,6),
  (0,2,4,5,1,3,6), (0,3,1,2,4,5,6),
  (0,3,1,5,4,2,6), (0,4,2,1,3,5,6),
  (0,4,2,1,5,3,6), (0,4,5,3,1,2,6).
a(7) = 3 due to the permutations(0,1,4,3,2,5,6,7), (0,1,6,5,2,3,4,7), (0,5,2,3,4,1,6,7).
a(8) > 0 due to the permutation (0,1,4,7,3,2,6,5,8).
a(9) > 0 due to the permutation (0,1,2,5,4,7,6,3,8,9).
a(10) > 0 due to the permutation (0,1,3,2,5,8,6,4,9,7,10).

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.

Cf. A228860, A228917, A228956.

(*A program to compute the 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]]+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!}]

a(11)-a(25) from Max Alekseyev, Sep 13 2013

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.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Sep 10 2013

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

			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).

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

Cf. A000040, A051252, A228917, A228956, A228886.

(* 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!}]

a(10)-a(18) from Alois P. Heinz, Sep 10 2013

A229082 Number of circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that all 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 of the form (p-1)/2 with p an odd prime.

Original entry on oeis.org

1, 1, 1, 0, 2, 3, 7, 11, 9, 5, 41, 82, 254, 2412, 9524, 13925, 85318, 220818, 1662421, 10496784, 20690118, 97200566, 460358077

Offset: 1

Zhi-Wei Sun, Sep 13 2013

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

Note that if a circular permutation i_0, i_1, ..., i_n of 0, 1, ..., n with i_0 = 0 meets the requirement then we must have i_n = 1. This can be explained as follows: If i_n > 1, then 3 | i_n since 2*(i_n^2+0)+1 is a prime not divisible by 3, and similarly i_{n-1},...,i_1 are also multiples of 3 since 2*(i_{n-1}^2+i_n)+1, ..., 2*(i_1^2+i_2)+1 are primes not divisible by 3. Therefore, i_n > 1 would lead to a contradiction.

			a(1) = 1 due to the circular permutation (0,1).
a(2) = 1 due to the circular permutation (0,2,1).
a(3) = 1 due to the circular permutation (0,3,2,1).
a(5) = 2 due to the circular permutations
   (0,3,2,4,5,1) and (0,3,5,4,2,1).
a(6) = 3 due to the circular permutations
   (0,3,6,5,4,2,1), (0,6,3,2,4,5,1), (0,6,3,5,4,2,1).
a(7) = 7 due to the circular permutations
   (0,3,6,5,4,2,7,1), (0,3,6,5,4,7,2,1), (0,6,3,2,4,7,5,1),
   (0,6,3,2,5,4,7,1), (0,6,3,2,7,4,5,1), (0,6,3,5,4,2,7,1),
   (0,6,3,5,4,7,2,1).
a(8) = 11 due to the circular permutations
   (0,3,6,5,8,4,2,7,1), (0,3,6,5,8,4,7,2,1),
   (0,3,6,8,4,2,7,5,1), (0,4,6,8,4,7,2,5,1),
   (0,3,6,8,5,4,2,7,1), (0,3,6,8,5,4,7,2,1),
   (0,6,3,2,4,7,5,8,1), (0,6,3,2,5,8,4,7,1),
   (0,6,3,2,7,4,5,8,1), (0,6,3,5,8,4,2,7,1),
   (0,6,3,5,8,4,7,2,1).
a(9) = 9 due to the circular permutations
   (0,6,3,9,2,4,7,5,8,1), (0,6,3,9,2,5,8,4,7,1),
   (0,6,3,9,2,7,4,5,8,1), (0,6,3,9,5,8,4,2,7,1),
   (0,6,3,9,5,8,4,7,2,1), (0,6,3,9,8,4,2,7,5,1),
   (0,6,3,9,8,4,7,2,5,1), (0,6,3,9,8,5,4,2,7,1),
   (0,6,3,9,8,5,4,7,2,1).
a(20) > 0 due to the circular permutation
  (0,3,12,9,15,18,6,20,19,14,13,4,2,7,16,17,11,10,5,8,1).

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

Cf. A228917, A228956, A229005, A229038.

(* A program to compute required circular permutations for n = 7. *)
p[i_,j_]:=tp[i,j]=PrimeQ[2(i^2+j)+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!}]

a(10)-a(23) from Alois P. Heinz, Sep 13 2013

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.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Sep 15 2013

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(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.

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.

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

(* 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!}]

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

A229232 Number of undirected circular permutations pi(1), ..., pi(n) of 1, ..., n with the n numbers pi(1)pi(2)-1, pi(2)pi(3)-1, ..., pi(n-1)pi(n)-1, pi(n)pi(1)-1 all prime.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 2, 2, 8, 2, 241, 0, 693, 376, 7687, 1082, 127563, 25113, 1353842, 559649

Offset: 1

Zhi-Wei Sun, Sep 16 2013

Conjecture: a(n) > 0 for all n > 5 with n not equal to 13.
Zhi-Wei Sun also made the following conjectures:
(1) For any integer n > 1, there is a permutation pi(1), ..., pi(n) of 1, ..., n such that the n numbers 2*pi(1)*pi(2)-1, ..., 2*pi(n-1)*pi(n)-1, 2*pi(n)*pi(1)-1 are all prime. Also, for any positive integer n not equal to 4, there is a permutation pi(1), ..., pi(n) of 1, ..., n such that the n numbers 2*pi(1)*pi(2)+1, ..., 2*pi(n-1)*pi(n)+1, 2*pi(n)*pi(1)+1 are all prime.
(2) Let F be a finite field with q > 7 elements. Then, there is a circular permutation a_1,...,a_{q-1} of the q-1 nonzero elements of F such that all the q-1 elements a_1*a_2-1, a_2*a_3-1, ..., a_{q-2}*a_{q-1}-1, a_{q-1}*a_1-1 are primitive elements of the field F (i.e., generators of the multiplicative group F\{0}). Also, there is a circular permutation b_1,...,b_{q-1} of the q-1 nonzero elements of F such that all the q-1 elements b_1*b_2+1, b_2*b_3+1, ..., b_{q-2}*b_{q-1}+1, b_{q-1}*b_1+1 are primitive elements of the field F.

			a(4) = 1 due to the circular permutation (1,3,2,4).
a(6) = 2 due to the circular permutations
   (1,3,2,4,5,6) and (1,3,2,6,5,4).
a(7) = 1 due to the circular permutation (1,3,2,7,6,5,4).
a(8) = 2 due to the circular permutations
   (1,3,2,7,6,5,4,8) and (1,4,5,6,7,2,3,8).
a(9) = 2 due to the circular permutations
   (1,3,4,5,6,7,2,9,8) and (1,3,8,9,2,7,6,5,4).
a(10) = 8 due to the circular permutations
   (1,3,4,5,6,7,2,9,10,8), (1,3,4,5,6,7,2,10,9,8),
   (1,3,8,9,10,2,7,6,5,4), (1,3,8,10,9,2,7,6,5,4),
   (1,3,10,8,9,2,7,6,5,4), (1,3,10,9,2,7,6,5,4,8),
   (1,4,5,6,7,2,3,10,9,8), (1,4,5,6,7,2,9,10,3,8).
a(13) = 0 since 8 is the unique j among 1, ..., 12 with 13*j-1 prime.

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

Cf. A051252, A227456, A228917, A228956, A229082.

(* A program to compute required circular permutations for n = 8. To get "undirected" circular permutations, we should identify a circular permutation with the one of the opposite direction; for example, (1,8,4,5,6,7,2,3) is identical to (1,3,2,7,6,5,4,8) if we ignore direction. Thus, a(8) is half of the number of circular permutations yielded by this program. *)
V[i_]:=V[i]=Part[Permutations[{2,3,4,5,6,7,8}],i]
f[i_,j_]:=f[i,j]=PrimeQ[i*j-1]
m=0
Do[Do[If[f[If[j==0,1,Part[V[i],j]],If[j<7,Part[V[i],j+1],1]]==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!}]

a(11)-a(21) from Pontus von Brömssen, Jan 08 2025

A229543 Number of undirected circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that all the n+1 adjacent distances |i_0-i_1|, |i_1-i_2|, ..., |i_{n-1}-i_n|, |i_n-i_0| are perfect squares.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 9, 14, 32, 184, 123, 696, 935, 6554, 21105, 60756, 241780, 517970, 1835109, 5741024, 16091004, 63590090, 285113492, 1098219807

Offset: 1

Zhi-Wei Sun, Sep 25 2013

Theorem: For any integer n > 5, there is a circular permutation i_0, i_1, ..., i_n of 0, 1, ..., n with both 1 and 4 adjacent to 0 such that all the n+1 adjacent distances |i_0-i_1|, |i_1-i_2|, ..., |i_{n-1}-i_n|, |i_n-i_0| are perfect squares.
Proof: By examples, the result holds for n = 6, ..., 11. Below we assume n > 11  and exhibit a circular permutation meeting the requirement.
  If n == 0 (mod 6), then we may take the circular permutation (n,n-4,n-5,n-6,n-10,n-11,n-12,...,8,7,6,2,3,4,0,1,5,9,10,11,...,n-9,n-8,n-7,n-3,n-2,n-1).
  If n == 3 (mod 6), then we may take the circular permutation
  (n,n-4,n-5,n-6,n-10,n-11,n-12,...,11,10,9,5,1,0,4,3,2,6,7,8,...,n-9,n-8,n-7,n-3,n-2,n-1).
  If n == 1 (mod 6), then we may take the circular permutation
  (n,n-4,n-5,n-6,n-10,n-11,n-12,...,9,8,7,3,2,1,0,4,5,6,10,11,12,...,n-9,n-8,n-7,n-3,n-2,n-1).
  If n == 4 (mod 6), then we may take the circular permutation (n,n-4,n-5,n-6,n-10,n-11,n-12,...,12,11,10,6,5,4,0,1,2,3,7,8,9,...,n-9,n-8,n-7,n-3,n-2,n-1).
  If n == 2 (mod 6), then we may take the circular permutation (n,n-4,n-5,n-6,n-10,n-11,n-12,...,10,9,8,4,0,1,5,6,2,3,7,11,12,13,...,n-9,n-8,n-7,n-3,n-2,n-1).
  If n == 5 (mod 6), then we may take the circular permutation
  (n,n-4,n-5,n-6,n-10,n-11,n-12,...,13,12,11,7,3,2,6,5,1,0,4,8,9,10,...,n-9,n-8,n-7,n-3,n-2,n-1).
Zhi-Wei Sun also used a similar method to show that for any positive integer n not equal to 2 or 4 there is a circular permutation i_0, i_1, ..., i_n of 0, 1, ..., n such that all the n+1 adjacent distances |i_0-i_1|, |i_1-i_2|, ..., |i_{n-1}-i_n|, |i_n-i_0| are triangular numbers.

			a(1) = 1 due to the circular permutation (0,1).
a(4) = 1 due to the circular permutation (0,1,2,3,4).
a(6) = 1 due to the circular permutation (0,1,5,6,2,3,4).
a(7) = 1 due to the circular permutation (0,1,2,3,7,6,5,4).
a(8) = 1 due to the circular permutation (0,1,5,6,2,3,7,8,4).
a(9) = 9 due to the circular permutations
  (0,1,2,3,4,5,6,7,8,9), (0,1,2,3,4,8,7,6,5,9),
  (0,1,2,3,7,6,5,4,8,9), (0,1,2,3,7,6,5,9,8,4),
  (0,1,2,6,5,4,3,7,8,9), (0,1,2,6,5,9,8,7,3,4),
  (0,1,5,4,3,2,6,7,8,9), (0,1,5,9,8,7,6,2,3,4),
  (0,4,3,2,1,5,6,7,8,9).
a(10) > 0 due to the permutation (0,1,2,3,7,8,9,10,6,5,4).
a(11) > 0 due to the permutation (0,1,5,9,8,7,11,10,6,2,3,4).

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

Cf. A000290, A185645, A228626, A228728, A228956, A229005.

(* A program to compute required circular permutations for n = 8. To get "undirected" circular permutations, we should identify a circular permutation with the one of the opposite direction. Thus a(8) is half of the number of circular permutations yielded by this program. *)
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
f[i_,j_]:=f[i,j]=SQ[Abs[i-j]]
V[i_]:=V[i]=Part[Permutations[{1,2,3,4,5,6,7,8}],i]
m=0
Do[Do[If[f[If[j==0,0,Part[V[i],j]],If[j<8,Part[V[i],j+1],0]]==False,Goto[aa]],{j,0,8}];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]," ",Part[V[i],8]];Label[aa];Continue,{i,1,8!}]

a(10)-a(24) from Alois P. Heinz, Sep 26 2013

a(25)-a(26) from Max Alekseyev, Jan 08 2015

cp's OEIS Frontend

A228886 Number of permutations i_0,i_1,...,i_n of 0,1,...,n with i_0 = 0 and i_n = n, and with i_0+i_1, i_1+i_2, ..., i_{n-1}+i_n, i_n+i_0 all coprime to both n-1 and n+1.

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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.

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A229082 Number of circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that all 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 of the form (p-1)/2 with p an odd prime.

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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.

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A229232 Number of undirected circular permutations pi(1), ..., pi(n) of 1, ..., n with the n numbers pi(1)*pi(2)-1, pi(2)*pi(3)-1, ..., pi(n-1)*pi(n)-1, pi(n)*pi(1)-1 all prime.

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A229543 Number of undirected circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that all the n+1 adjacent distances |i_0-i_1|, |i_1-i_2|, ..., |i_{n-1}-i_n|, |i_n-i_0| are perfect squares.

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A229232 Number of undirected circular permutations pi(1), ..., pi(n) of 1, ..., n with the n numbers pi(1)pi(2)-1, pi(2)pi(3)-1, ..., pi(n-1)pi(n)-1, pi(n)pi(1)-1 all prime.