A071059 - OEIS

%I #24 Sep 21 2023 14:16:38

%S 1,1,1,1,1,2,2,6,4,7,6,11,11,53,53,181,171,939,925,4432,4545,15811,

%T 15583,67891,68193,434963,388975,2718150,3113343,21580655,18425145,

%U 92370364,94887088,564878656,572364768,4545704064,4092294083,36878092180,36363930614

%N Number of ways of pairing even numbers in the range 1 to n with odd numbers in the range n+1 to 2n such that each pair sums to a prime.

%H Martin Fuller, <a href="/A071059/b071059.txt">Table of n, a(n) for n = 1..70</a>

%F a(2n) = A071058(2n).

%e a(6)=2 because there are two ways: 2+9, 4+7, 6+11 and 2+11, 4+9, 6+7.

%p f:= proc(n) local m;

%p   m:= floor(n/2);

%p  LinearAlgebra:-Permanent(Matrix(m,m,

%p     (i,j) -> `if`(isprime((i+j-2)*2 + n + 3 + (n mod 2)),1,0)))

%p end proc:

%p f(1):= 1:

%p map(f, [$1..40]); # _Robert Israel_, Sep 21 2023

%t a[n_] := a[n] = If[n == 1, 1, Module[{s1, s2, s3, s4, i, ik, km},

%t s1 = Select[Flatten[Outer[List, Range[2, n, 2], Range[2n-1, n+1, -2]], 1], PrimeQ[Total[#]]&];

%t s2 = SplitBy[s1, First];

%t km = Length[s2];

%t ik = Table[{i[k], 1, Length[s2[[k]]]}, {k, 1, km}];

%t s3 = Table[Table[s2[[k, i[k]]], {k, 1, km}], Evaluate[Sequence @@ ik]] // Flatten[#, km - 1]&;

%t s4 = Select[s3, Length[Union[Flatten[#]]] == 2km&];

%t s4 // Length]];

%t Table[Print[n, " ", a[n]]; a[n], {n, 1, 20}] (* _Jean-François Alcover_, Aug 10 2022 *)

%o (PARI) a(n)=matpermanent(matrix(n\2,n\2,i,j,isprime((i+j-2)*2+n+3+(n%2)))); \\ _Martin Fuller_, Sep 21 2023

%Y The product of this sequence and A071058 gives A070897.

%K nice,nonn

%O 1,6

%A _T. D. Noe_, May 25 2002

%E More terms from _David W. Wilson_, May 27 2002

%E a(31)-a(37) from _Donovan Johnson_, Aug 12 2010