OFFSET
1,6
COMMENTS
See a similar problem, but for the set of numbers {0 through (n-1)}. - Stanislav Sykora, May 30 2014
LINKS
Gary Antonick, Numberplay: Bernardo Recamán’s Primes in a Circle Puzzle, Jun 17 2013.
Stanislav Sykora, On Neighbor-Property Cycles, Stan's Library, Volume V, 2014; Table III.
EXAMPLE
For n = 6 the a(6) = 2 solutions are (1, 4, 9, 2, 5, 12, 7, 10, 3, 8, 11, 6) and (1, 6, 11, 8, 3, 10, 7, 4, 9, 2, 5, 12) because abs(1 - 4) = 3 and 1 + 4 = 5 are prime, etc.
MATHEMATICA
A227050[n_] :=
Count[Map[lpf, Map[j1f, Permutations[Range[2, 2 n]]]], 0]/2;
j1f[x_] := Join[{1}, x, {1}];
lpf[x_] := Length[
Join[Select[asf[x], ! PrimeQ[#] &],
Select[Differences[x], ! PrimeQ[#] &]]];
asf[x_] := Module[{i}, Table[x[[i]] + x[[i + 1]], {i, Length[x] - 1}]];
Table[A227050[n], {n, 1, 6}]
(* OR, a less simple, but more efficient implementation. *)
A227050[n_, perm_, remain_] := Module[{opt, lr, i, new},
If[remain == {},
If[PrimeQ[First[perm] - Last[perm]] &&
PrimeQ[First[perm] + Last[perm]], ct++];
Return[ct],
opt = remain; lr = Length[remain];
For[i = 1, i <= lr, i++,
new = First[opt]; opt = Rest[opt];
If[! (PrimeQ[Last[perm] - new] && PrimeQ[Last[perm] + new]),
Continue[]];
A227050[n, Join[perm, {new}],
Complement[Range[2 n], perm, {new}]];
];
Return[ct];
];
];
Table[ct = 0; A227050[n, {1}, Range[2, 2 n]]/2, {n, 1, 10}]
(* Robert Price, Oct 22 2018 *)
PROG
(C++) // Listed in the Sykora link.
CROSSREFS
KEYWORD
nonn,more,hard
AUTHOR
Tim Cieplowski, Jun 29 2013
EXTENSIONS
a(15)-a(18) added by Tim Cieplowski, Jan 04 2015
a(19) from Fausto A. C. Cariboni, Jun 06 2017
a(20) from Bert Dobbelaere, Feb 15 2020
a(21) from Martin Fuller, Mar 11 2026
STATUS
approved