A228886 -id:A228886 - cp's OEIS frontend

A228917 Number of undirected circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that i_0+i_1, i_1+i_2, ...,i_{n-1}+i_n, i_n+i_0 are among those k with 6k-1 and 6k+1 twin primes.

1, 1, 1, 2, 2, 2, 5, 2, 12, 39, 98, 526, 2117, 6663, 15043, 68403, 791581, 4826577, 19592777, 102551299, 739788968, 4449585790, 36547266589, 324446266072, 2743681178070

Offset: 1

Zhi-Wei Sun, Sep 08 2013

Conjecture: a(n) > 0 for all n > 0.
This implies the twin prime conjecture, and it is similar to the prime circle problem mentioned in A051252.
For each n = 2,3,... construct an undirected simple graph T(n) with vertices 0,1,...,n which has an edge connecting two distinct vertices i and j if and only if 6*(i+j)-1 and 6*(i+j)+1 are twin primes. Then a(n) is just the number of Hamiltonian cycles contained in T(n). Thus a(n) > 0 if and only if T(n) is a Hamilton graph.
Zhi-Wei Sun also made the following similar conjectures for odd primes, Sophie Germain primes, cousin primes and sexy primes:
(1) For any integer n > 0, there is a permutation i_0, i_1, ..., i_n of 0, 1, ..., n such that i_0+i_1, i_1+i_2, ..., i_{n-1}+i_n, i_n+i_0 are integers of the form (p-1)/2, where p is an odd prime. Also, we may replace the above (p-1)/2 by (p+1)/4 or (p-1)/6; when n > 4 we may substitute (p-1)/4 for (p-1)/2.
(2) For any integer n > 2, there is a permutation i_0, i_1, ..., i_n of 0, 1,..., n  such that i_0+i_1, i_1+i_2, ..., i_{n-1}+i_n, i_n+i_0 are integers of the form (p+1)/6, where p is a Sophie Germain prime.
(3) For any integer n > 3, there is a permutation i_0, i_1, ..., i_n of 0, 1,..., n  such that i_0+i_1, i_1+i_2, ..., i_{n-1}+i_n, i_n+i_0 are among those integers k with 6*k+1 and 6*k+5 both prime.
(4) For any integer n > 4, there is a permutation i_0, i_1, ..., i_n of 0, 1,..., n  such that i_0+i_1, i_1+i_2, ..., i_{n-1}+i_n, i_n+i_0 are among those integers k with 2*k-3 and 2*k+3 both prime.

			a(n) = 1 for n = 1,2,3 due to the permutation (0,...,n).
a(4) = 2 due to the permutations (0,1,4,3,2) and (0,2,1,4,3).
a(5) = 2 due to the permutations (0,1,4,3,2,5), (0,3,4,1,2,5).
a(6) = 2 due to the permutations
  (0,1,6,4,3,2,5) and (0,3,4,6,1,2,5).
a(7) = 5 due to the permutations
  (0,1,6,4,3,2,5,7), (0,1,6,4,3,7,5,2), (0,2,1,6,4,3,7,5),
  (0,3,4,6,1,2,5,7), (0,5,2,1,6,4,3,7).
a(8) = 2 due to the permutations
  (0,1,6,4,8,2,3,7,5) and (0,1,6,4,8,2,5,7,3).
a(9) = 12 due to the permutations
  (0,1,6,4,3,9,8,2,5,7), (0,1,6,4,8,9,3,2,5,7),
  (0,1,6,4,8,9,3,7,5,2), (0,2,1,6,4,8,9,3,7,5),
  (0,2,8,9,1,6,4,3,7,5), (0,3,4,6,1,9,8,2,5,7),
  (0,3,9,1,6,4,8,2,5,7), (0,3,9,8,4,6,1,2,5,7),
  (0,5,2,1,6,4,8,9,3,7), (0,5,2,8,4,6,1,9,3,7),
  (0,5,2,8,9,1,6,4,3,7), (0,5,7,3,9,1,6,4,8,2).
a(10) > 0 due to the permutation (0,5,2,3,9,1,6,4,8,10,7).
a(11) > 0 due to the permutation (0,10,8,9,3,7,11,6,4,1,2,5).
a(12) > 0 due to the permutation
        (0, 5, 2, 1, 6, 4, 3, 9, 8, 10, 7, 11, 12).

Zhi-Wei Sun, Twin primes and circular permutations, a message to Number Theory List, Sept. 8, 2013.
Z.-W. Sun, Some new problems in additive combinatorics, arXiv preprint arXiv:1309.1679 [math.NT], 2013-2014.

Cf. A000040, A001359, A006512, A023200, A046132, A023201, A046117, A005384, A051252, A228766, A228860, 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, (0,7,5,2,3,4,6,1) is identical to (0,1,6,4,3,2,5,7) if we ignore direction. Thus a(7) is half of the number of circular permutations yielded by this program. *)
tp[n_]:=tp[n]=PrimeQ[6n-1]&&PrimeQ[6n+1]
V[i_]:=Part[Permutations[{1,2,3,4,5,6,7}],i]
m=0
Do[Do[If[tp[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(25) from Max Alekseyev, Sep 12 2013

A228956 Number of undirected circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that all the 2*n+2 numbers |i_0 +/- i_1|, |i_1 +/- i_2|, ..., |i_{n-1} +/- i_n|, |i_n +/- i_0| have the form (p-1)/2 with p an odd prime.

Original entry on oeis.org

1, 1, 1, 1, 5, 9, 17, 84, 30, 127, 791, 2404, 11454, 27680, 25942, 137272, 515947, 2834056, 26583034, 82099932, 306004652, 4518630225, 11242369312, 8942966426, 95473633156, 533328765065

Offset: 1

Zhi-Wei Sun, Sep 09 2013

Conjecture: a(n) > 0 for all n > 0.
Note that if i-j = (p-1)/2 and i+j = (q-1)/2 for some odd primes p and q then 4*i+2 is the sum of the two primes p and q. So the conjecture is related to Goldbach's conjecture.
Zhi-Wei Sun also made the following similar conjecture:  For any integer n > 5, there exists a circular permutation i_0, i_1, ..., i_n of 0, 1, ..., n such that all the 2*n+2 numbers 2*|i_k-i_{k+1}|+1 and  2*(i_k+i_{k+1})-1 (k = 0,...,n) (with i_{n+1} = i_0) are primes.

			a(n) = 1 for n = 1,2,3 due to the natural circular permutation (0,...,n).
a(4) = 1 due to the circular permutation (0,1,4,2,3).
a(5) = 5 due to the circular permutations (0,1,2,4,5,3), (0,1,4,2,3,5), (0,1,4,5,3,2), (0,2,1,4,5,3), (0,3,2,1,4,5).
a(6) = 9 due to the circular permutations
  (0,1,2,4,5,3,6), (0,1,2,4,5,6,3), (0,1,4,2,3,5,6),
  (0,1,4,2,3,6,5), (0,1,4,5,6,3,2), (0,2,1,4,5,3,6),
  (0,2,1,4,5,6,3), (0,3,2,1,4,5,6), (0,5,4,1,2,3,6).
a(7) = 17 due to the circular permutations
  (0,1,2,7,4,5,3,6), (0,1,2,7,4,5,6,3), (0,1,4,7,2,3,5,6),
  (0,1,4,7,2,3,6,5), (0,1,7,2,4,5,3,6), (0,1,7,2,4,5,6,3),
  (0,1,7,4,2,3,5,6), (0,1,7,4,2,3,6,5), (0,1,7,4,5,6,3,2),
  (0,2,1,7,4,5,3,6), (0,2,1,7,4,5,6,3), (0,2,7,1,4,5,3,6),
  (0,2,7,1,4,5,6,3), (0,3,2,1,7,4,5,6), (0,3,2,7,1,4,5,6),
  (0,5,4,1,7,2,3,6), (0,5,4,7,1,2,3,6).

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

Cf. A000040, A051252, A002375, A228917, 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, (0,6,3,5,4,7,2,1) is identical to (0,1,2,7,4,5,3,6) 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-j]+1]&&PrimeQ[2(i+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(26) from Max Alekseyev, Sep 17 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

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

cp's OEIS Frontend

A228917 Number of undirected circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that i_0+i_1, i_1+i_2, ...,i_{n-1}+i_n, i_n+i_0 are among those k with 6*k-1 and 6*k+1 twin primes.

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A228956 Number of undirected circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that all the 2*n+2 numbers |i_0 +/- i_1|, |i_1 +/- i_2|, ..., |i_{n-1} +/- i_n|, |i_n +/- i_0| have the form (p-1)/2 with p an odd prime.

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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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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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A228917 Number of undirected circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that i_0+i_1, i_1+i_2, ...,i_{n-1}+i_n, i_n+i_0 are among those k with 6k-1 and 6k+1 twin primes.