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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 6*k-1 and 6*k+1 twin primes. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..25.

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, 2013

EXAMPLE

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).

MATHEMATICA

(* 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!}]

CROSSREFS

Cf. A000040, A001359, A006512, A023200, A046132, A023201, A046117, A005384, A051252, A228766, A228860, A228886.

Sequence in context: A077913 A069862 A075002 * A061311 A174960 A210239

Adjacent sequences:  A228914 A228915 A228916 * A228918 A228919 A228920

KEYWORD

nonn,more

AUTHOR

Zhi-Wei Sun, Sep 08 2013

EXTENSIONS

a(10)-a(25) from Max Alekseyev, Sep 12 2013

STATUS

approved

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Last modified May 26 19:44 EDT 2019. Contains 323597 sequences. (Running on oeis4.)