A242528 - OEIS

%I #16 Oct 23 2018 03:03:02

%S 0,0,0,0,0,0,0,0,0,0,0,2,4,18,13,62,8,133,225,209,32,2644,4462,61341,

%T 113986,750294,176301,7575912,3575686,7705362,36777080,108638048,

%U 97295807

%N Number of cyclic arrangements of {0,1,...,n-1} such that both the difference and the sum of any two neighbors are prime.

%C a(n)=NPC(n;S;P) is the count of all neighbor-property cycles for a specific set S of n elements and a specific pair-property P. For more details, see the link and A242519.

%C In this case the set is S={0 through n-1}. For the same pair-property P but the set S={1 through n}, see A227050.

%H 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 For n=12 (the first n for which a(n)>0) there are two such cycles:

%e C_1={0, 5, 2, 9, 4, 1, 6, 11, 8, 3, 10, 7},

%e C_2={0, 7, 10, 3, 8, 5, 2, 9, 4, 1, 6, 11}.

%t A242528[n_] :=

%t Count[Map[lpf, Map[j0f, Permutations[Range[n - 1]]]], 0]/2;

%t j0f[x_] := Join[{0}, x, {0}];

%t lpf[x_] := Length[

%t    Join[Select[asf[x], ! PrimeQ[#] &],

%t     Select[Differences[x], ! PrimeQ[#] &]]];

%t asf[x_] := Module[{i}, Table[x[[i]] + x[[i + 1]], {i, Length[x] - 1}]];

%t Table[A242528[n], {n, 1, 8}]

%t (* OR, a less simple, but more efficient implementation. *)

%t A242528[n_, perm_, remain_] := Module[{opt, lr, i, new},

%t    If[remain == {},

%t      If[PrimeQ[First[perm] - Last[perm]] &&

%t        PrimeQ[First[perm] + Last[perm]], ct++];

%t      Return[ct],

%t      opt = remain; lr = Length[remain];

%t      For[i = 1, i <= lr, i++,

%t       new = First[opt]; opt = Rest[opt];

%t       If[! (PrimeQ[Last[perm] - new] && PrimeQ[Last[perm] + new]),

%t        Continue[]];

%t       A242528[n, Join[perm, {new}],

%t        Complement[Range[n - 1], perm, {new}]];

%t       ];

%t      Return[ct];

%t      ];

%t    ];

%t Table[ct = 0; A242528[n, {0}, Range[n - 1]]/2, {n, 1, 18}]

%t (* _Robert Price_, Oct 22 2018 *)

%o (C++) See the link.

%Y Cf. A227050, A242519, A242520, A242521, A242522, A242523, A242524, A242525, A242526, A242527, A242529, A242530, A242531, A242532, A242533, A242534.

%K nonn,hard,more

%O 1,12

%A _Stanislav Sykora_, May 30 2014

%E a(29)-a(33) from _Fausto A. C. Cariboni_, May 20 2017