A070897
Number of ways of pairing numbers 1 to n with numbers n+1 to 2n such that each pair sums to a prime.
Original entry on oeis.org
1, 1, 1, 1, 2, 4, 8, 36, 40, 49, 126, 121, 440, 2809, 11395, 32761, 132183, 881721, 3015500, 19642624, 106493895, 249987721, 1257922092, 4609187881, 29262161844, 189192811369, 1068996265025, 7388339422500, 67416357342087, 465724670229025, 1979950199225010
Offset: 1
a(5)=2 because there are two ways: 1+10, 2+9, 3+8, 4+7, 6+5 and 1+6, 2+9, 3+10, 4+7, 5+8.
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import Data.List (permutations)
a070897 n = length $ filter (all ((== 1) . a010051))
$ map (zipWith (+) [1..n]) (permutations [n+1..2*n])
-- Reinhard Zumkeller, Mar 19 2011, Apr 16 2011 (fixed)
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<n ]& /@Select[ Range[ n+2, 3*n ], PrimeQ ], 1 ]; po=Position[ it, # ]&/@Range[ n ]; permoid=(Extract[ it, # ]-n)& /@(po /. {i_Integer, j_}->{i, 1} ); Length@Backtrack[ permoid, UnsameQ@@#&, Length[ # ]===n&, All ] ]; Noe/@Range[ 2, 16 ] (* from Wouter Meeussen *)
a[n_] := Permanent[Table[If[PrimeQ[i+j+n], 1, 0], {i, n}, {j, n}]]; Table[ an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 16}] (* Jean-François Alcover, Feb 26 2016 *)
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a(n)=my(a071058=matpermanent(matrix((n+1)\2,(n+1)\2,i,j,isprime((i+j-2)*2+n+3-(n%2))))); if(n%2==0, a071058^2, a071058*matpermanent(matrix(n\2,n\2,i,j,isprime((i+j-2)*2+n+3+(n%2))))); \\ Martin Fuller, Sep 21 2023
A071058
Number of ways of pairing odd numbers in the range 1 to n with even numbers in the range n+1 to 2n such that each pair sums to a prime.
Original entry on oeis.org
1, 1, 1, 1, 2, 2, 4, 6, 10, 7, 21, 11, 40, 53, 215, 181, 773, 939, 3260, 4432, 23431, 15811, 80724, 67891, 429108, 434963, 2748239, 2718150, 21654009, 21580655, 107459138, 92370364, 638616984, 564878656, 5055810584, 4545704064, 35787453599, 36878092180
Offset: 1
a(5)=2 because there are two ways: 1+10, 3+8, 6+5 and 1+6, 3+10, 5+8.
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a[n_] := a[n] = Module[{s1, s2, s3, s4, i, ik, km},
s1 = Select[Flatten[Outer[List, Range[1, n, 2], Range[2n, n+1, -2]], 1],
PrimeQ[Total[#]]&];
s2 = SplitBy[s1, First];
km = Length[s2];
ik = Table[{i[k], 1, Length[s2[[k]]]}, {k, 1, km}];
s3 = Table[Table[s2[[k, i[k]]], {k, 1, km}], Evaluate[Sequence @@ ik]] //
Flatten[#, km-1]&;
s4 = Select[s3, Length[Union[Flatten[#]]] == 2km&];
s4 // Length];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 20}] (* Jean-François Alcover, Aug 10 2022 *)
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a(n)=matpermanent(matrix((n+1)\2,(n+1)\2,i,j,isprime((i+j-2)*2+n+3-(n%2)))); \\ Martin Fuller, Sep 21 2023
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