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A070897
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Number of ways of pairing numbers 1 to n with numbers n+1 to 2n such that each pair sums to a prime.
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7
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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
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OFFSET
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1,5
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LINKS
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FORMULA
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a(n) = permanent(m), where the n X n matrix m is defined by m(i,j) = 1 or 0, depending on whether i+j+n is prime or composite, respectively. - T. D. Noe, Feb 10 2007
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EXAMPLE
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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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MATHEMATICA
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<<Combinatorica`; listQpart2[ n_ ] := {n-#, #}&/@Range[ Floor[ (n-1)/2 ] ]; Noe[ n_Integer ] := Module[ {it, permoid, po}, it=Union@Flatten[ Cases[ listQpart2[ # ], q_/; Max[ q ]<=2*n&&Max[ q ]>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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PROG
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(Haskell)
import Data.List (permutations)
a070897 n = length $ filter (all ((== 1) . a010051))
$ map (zipWith (+) [1..n]) (permutations [n+1..2*n])
(PARI) 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
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CROSSREFS
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KEYWORD
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nice,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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