

A185645


Number of permutations q_1,...,q_n of the first n primes p_1,...,p_n with q_1 = p_1 = 2 and q_n = p_n, and with q_1q_2, q_2q_3, ..., q_{n1}q_n, and q_nq_1 (if n>2) pairwise distinct.


10



1, 1, 1, 1, 3, 5, 10, 33, 153, 1060, 7337, 51434, 440728, 3587067, 28498105, 271208386, 3014400869, 35358507494
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OFFSET

1,5


COMMENTS

Conjecture: a(n) > 0 for all n > 0. In general, for any n consecutive primes p_k,...,p_{k+n1}, there always exists a permutation q_k,...,q_{k+n1} of p_k,...,p_{k+n1} with q_{k+n1} = p_{k+n1} such that the n1 numbers q_kq_{k+1}, q_{k+1}q_{k+2},...,q_{k+n2}q_{k+n1} are pairwise distinct. (In the case k = 2, this implies that a(n) > 0.)
Clearly there is no permutation a,b,c of 3,5,7 such that the three numbers ab,bc,ca are pairwise distinct. Also, for {a,b} = {7,11}, the three numbers 5a,ab,b13 cannot be pairwise distinct.
On Aug 31 2013, ZhiWei Sun proved the following extension of the general conjecture: Let a_1 < a_2 < ... < a_n be a sequence of n distinct real numbers in ascending order. Then there is a permutation b_1, ..., b_n of a_1, ..., a_n with b_n = a_n such that b_1b_2, b_2b_3, ..., b_{n1}b_n are pairwise distinct. In fact, when n = 2*k is even we may take (b_1,...,b_n) = (a_k,a_{k+1},a_{k1},a_{k+2},...,a_2,a_{2k1},a_1,a_{2k}); when n = 2*k1 is odd we may take (b_1,...,b_n) = (a_k,a_{k1},a_{k+1},a_{k2},a_{k+2},..., a_2,a_{2k2},a_1,a_{2k1}).
On Sep 01 2013, ZhiWei Sun made the following conjecture: (i) For any n distinct real numbers a_1, a_2, ..., a_n (not necessarily in ascending or descending order), there is a permutation b_1, ..., b_n of a_1, ..., a_n with b_1 = a_1 such that the n1 distances b_1b_2, b_2b_3, ..., b_{n1}b_n are pairwise distinct.
(ii) Let a_1, ..., a_n be n distinct elements of a finite additive abelian group G. Suppose that G is not divisible by n, or n is even and G is cyclic. Then there exists a permutation b_1, ..., b_n of a_1, ..., a_n with b_1 = a_1 such that the n1 differences b_{i+1}b_i (i = 1, ..., n1) are pairwise distinct.
We believe that part (ii) of the new conjecture holds at least when G is cyclic, and it might also hold when the group G is not abelian.
Note that if g is a primitive root modulo an odd prime p, then for any j = 0,...,p2 the permutation g^j, g^{j+1},...,g^{j+p2} of the p1 nonzero residues modulo p has adjacent differences g^{i+j+1}g^{i+j} = g^{i+j}*(g1) (i = 0, ..., p3) which are pairwise distinct modulo p.


LINKS

Table of n, a(n) for n=1..18.
Z.W. Sun, Some new problems in additive combinatorics, arXiv preprint arXiv:1309.1679 [math.NT], 20132014.


EXAMPLE

a(4) = 1 since (q_1,q_2,q_3,q_4) = (2,5,3,7) is the only suitable permutation.
a(5) = 3 since there are exactly three suitable permutations(q_1,q_2,q_3,q_4,q_5): (2,3,7,5,11), (2,5,7,3,11) and (2,7,3,5,11).
a(6) = 5 since there are exactly five suitable permutations (q_1,q_2,q_3,q_4,q_5,q_6): (2,5,3,11,7,13), (2,5,7,11,3,13), (2,7,5,11,3,13), (2,7,11,5,3,13), (2,11,5,7,3,13).
a(7) = 10, and the ten suitable permutations (q_1,...,q_7) are as follows:
(2,3,13,5,7,11,17), (2,7,3,13,11,5,17), (2,7,5,11,3,13,17),
(2,7,11,5,13,3,17), (2,11,3,13,7,5,17), (2,11,7,5,13,3,17),
(2,11,7,13,3,5,17), (2,11,7,13,5,3,17), (2,13,3,11,7,5,17),
(2,13,7,11,3,5,17).


MATHEMATICA

A185645[n_] := Module[{p, c = 0, i = 1, j, q},
If[n == 2, Return[1],
p = Permutations[Table[Prime[j], {j, 2, n  1}]];
While[i <= Length[p],
q = Join[{2}, p[[i]], {Prime[n]}]; i++;
If[Length[Union[Join[Table[Abs[q[[j]]  q[[j + 1]]], {j, 1, n  1}], {Abs[q[[n]]  q[[1]]]}]]] == n, c++]]; c]];
Table[A185645[n], {n, 1, 11}] (* Robert Price, Apr 04 2019 *)


CROSSREFS

Cf. A000040, A187815, A228626, A228728, A228762, A228766.
Sequence in context: A003186 A006826 A000214 * A060955 A317338 A305510
Adjacent sequences: A185642 A185643 A185644 * A185646 A185647 A185648


KEYWORD

nonn,more


AUTHOR

ZhiWei Sun, Aug 29 2013


EXTENSIONS

Name clarified by Robert Price, Apr 04 2019
a(12)a(18) from Bert Dobbelaere, Sep 08 2019


STATUS

approved



