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A242527 Number of cyclic arrangements (up to direction) of {0,1,...,n-1} such that the sum of any two neighbors is a prime.

%I A242527 #12 Oct 19 2018 16:58:53
%S A242527 0,0,0,0,1,1,2,6,6,22,80,504,840,6048,3888,37524,72976,961776,661016,
%T A242527 11533030,7544366,133552142,208815294,5469236592,6429567323,
%U A242527 153819905698,182409170334,4874589558919,7508950009102,209534365631599
%N A242527 Number of cyclic arrangements (up to direction) of {0,1,...,n-1} such that the sum of any two neighbors is a prime.
%C A242527 a(n)=NPC(n;S;P) is the count of all neighbor-property cycles for a specific set S={0,1,...,n-1} of n elements and a specific pair-property P. For more details, see the link and A242519.
%C A242527 For the same pair-property P but the set {1 through n}, see A051252. Using for pair-property the difference, rather than the sum, one obtains A228626.
%H A242527 S. Sykora, <a href="http://dx.doi.org/10.3247/SL5Math14.002">On Neighbor-Property Cycles</a>, <a href="http://ebyte.it/library/Library.html#math">Stan's Library</a>, Volume V, 2014.
%e A242527 The first such cycle is of length n=5: {0,2,1,4,3}.
%e A242527 The first case with 2 solutions is for cycle length n=7:
%e A242527 C_1={0,2,3,4,1,6,5}, C_2={0,2,5,6,1,4,3}.
%e A242527 The first and the last of the 22 such cycles of length n=10 are:
%e A242527 C_1={0,3,2,1,4,9,8,5,6,7}, C_22={0,5,8,9,4,3,2,1,6,7}.
%t A242527 A242527[n_] := Count[Map[lpf, Map[j0f, Permutations[Range[n - 1]]]], 0]/2;
%t A242527 j0f[x_] := Join[{0}, x, {0}];
%t A242527 lpf[x_] := Length[Select[asf[x], ! PrimeQ[#] &]];
%t A242527 asf[x_] := Module[{i}, Table[x[[i]] + x[[i + 1]], {i, Length[x] - 1}]];
%t A242527 Table[A242527[n], {n, 1, 10}]
%t A242527 (* OR, a less simple, but more efficient implementation. *)
%t A242527 A242527[n_, perm_, remain_] := Module[{opt, lr, i, new},
%t A242527    If[remain == {},
%t A242527      If[PrimeQ[First[perm] + Last[perm]], ct++];
%t A242527      Return[ct],
%t A242527      opt = remain; lr = Length[remain];
%t A242527      For[i = 1, i <= lr, i++,
%t A242527       new = First[opt]; opt = Rest[opt];
%t A242527       If[! PrimeQ[Last[perm] + new], Continue[]];
%t A242527       A242527[n, Join[perm, {new}],
%t A242527        Complement[Range[n - 1], perm, {new}]];
%t A242527       ];
%t A242527      Return[ct];
%t A242527      ];
%t A242527    ];
%t A242527 Table[ct = 0; A242527[n, {0}, Range[n - 1]]/2, {n, 1, 15}]
%t A242527 (* _Robert Price_, Oct 18 2018 *)
%o A242527 (C++) See the link.
%Y A242527 Cf. A051252, A228626, A242519, A242520, A242521, A242522, A242523, A242524, A242525, A242526, A242528, A242529, A242530, A242531, A242532, A242533, A242534.
%K A242527 nonn,hard
%O A242527 1,7
%A A242527 _Stanislav Sykora_, May 30 2014
%E A242527 a(23)-a(30) from _Max Alekseyev_, Jul 09 2014