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 A227050 Number of essentially different ways of arranging numbers 1 through 2n around a circle so that the sum and absolute difference of each pair of adjacent numbers are prime. 5
 0, 0, 0, 0, 0, 2, 1, 4, 88, 0, 976, 22277, 22365, 376002, 3172018, 5821944, 10222624, 424452210, 6129894510, 38164752224 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS See a similar problem, but for the set of numbers {0 through (n-1)}. - Stanislav Sykora, May 30 2014 LINKS Table of n, a(n) for n=1..20. 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 Cf. similar sequences: A051252 (with sums of neighbors prime), A242527 (with sums of neighbors prime), A228626 (with differences of neighbors prime), A242528 (with sums and differences of neighbors prime). Sequence in context: A009830 A053374 A355517 * A093876 A322334 A198371 Adjacent sequences: A227047 A227048 A227049 * A227051 A227052 A227053 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 STATUS approved

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Last modified August 14 09:55 EDT 2024. Contains 375159 sequences. (Running on oeis4.)