%I #6 Mar 31 2012 14:40:08
%S 2,4,26,112,365,449,453
%N Numbers n such that (2n)!-(2n-2)!+(2n-4)!-...+(-1)^n 0! is prime.
%C There is no further term up to 1000. Consider that 3 divides n!+(n-1)!+(n-2)!+...+1! (n > 1), so this number is composite for n > 2. Also 5 divides n!-(n-1)!+...+(-1)^n*1! for n > 2, so this number is composite for n > 3.
%e 4 is in the sequence because 8!-6!+4!-2!+1 =39623 is prime.
%t Do[If[PrimeQ[Sum[(-1)^(n-k)(2k)!, {k, 0, n}]], Print[n], {n, 1000}]
%Y Cf. A001272, A002981, A002982.
%K more,nonn
%O 1,1
%A _Farideh Firoozbakht_, Jul 18 2003
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