%I #23 Apr 04 2019 14:33:43

%S 10,2,7,4,10,17,15,15,17,11,4,23,33,24,19,16,24,16,31,39,39,30,24,11,

%T 15,39,30,52,66,41,29,23,48,43,15,15,43,48,39,30,30,52,68,64,68,34,19,

%U 27,39,35,22,36,32,20,19,32,38,72,71,59

%N Number of permutations q_1, ..., q_7 of the 7 consecutive primes p_n, p_{n+1}, ..., p_{n+6} with q_1 = p_n and q_7 = p_{n+6}, and with |q_1-q_2|, |q_2-q_3|, ..., |q_6-q_7|, |q_7-q_1| pairwise distinct, where p_k denotes the k-th prime.

%C For each k = 3,4,5,6 there are k consecutive primes p_n, p_{n+1}, ..., p_{n+k-1} such that there is no permutation q_1, ..., q_k of p_n, p_{n+1}, ..., p_{n+k-1} with |q_1-q_2|, ..., |q_{k-1}-q_k|, |q_k-q_1| pairwise distinct. Such consecutive primes include (3, 5, 7), (5, 7, 11, 13), (3, 5, 7, 11, 13), and (p_{2209}, p_{2210}, ..., p_{2214}) = (19471, 19477, 19483, 19489, 19501, 19507).

%C For k > 7 the author once thought that for any k consecutive primes p_n, p_{n+1}, ..., p_{n+k-1} there always exists a permutation q_1, ..., q_k of p_n, p_{n+1}, ..., p_{n+k-1} with |q_1-q_2|, ..., |q_{k-1}-q_k|, |q_k-q_1| pairwise distinct. But this is unlikely to be true as pointed out by _Noam D. Elkies_.

%C See also A185645 for a related conjecture.

%H Zhi-Wei Sun, <a href="/A187815/b187815.txt">Table of n, a(n) for n = 1..10000</a>

%H Noam D. Elkies, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;fdca3ef0.1308">Re: A conjecture on permutations of consecutive primes</a>, a message to Number Theory List, August 31, 2013.

%e a(2) = 2 since there are exactly two permutations q_1,...,q_7 of 3,5,7,11,13,17,19 meeting the requirement: (q_1,...,q_7) = (3, 7, 17, 11, 13, 5, 19), (3, 11, 13, 7, 17, 5, 19).

%t V[n_,i_]:=Part[Permutations[{Prime[n+1],Prime[n+2],Prime[n+3],Prime[n+4],Prime[n+5]}],i]

%t Do[m=0;Do[If[Length[Union[{Abs[Part[V[n,i],1]-Prime[n]]},Table[Abs[Part[V[n,i],j]-If[j<5,Part[V[n,i],j+1],Prime[n+6]]],{j,1,5}]]]<6,Goto[aa]];

%t m=m+1;Label[aa];Continue,{i,1,5!}];Print[n," ",m];Continue,{n,1,20}]

%t A187815[n_] := Module[{p, c = 0, i = 1, j, q},

%t p = Permutations[Table[Prime[j], {j, n + 1, n + 5}]];

%t While[i <= Length[p],

%t q = Join[{Prime[n]}, p[[i]], {Prime[n + 6]}]; i++;

%t If[Length[

%t Union[Join[

%t Table[Abs[q[[j]] - q[[j + 1]]], {j, 1, 6}], {Abs[

%t q[[7]] - q[[1]]]}]]] == 7, c++]]; c];

%t Table[A187815[n], {n, 1, 60}] (* _Robert Price_, Apr 04 2019 *)

%Y Cf. A000040, A185645, A228728.

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, Aug 30 2013