%I #9 Jul 18 2019 14:29:05
%S 5,317,1217216458429656137
%N Primes of the form [k!! - (k-1)!! + (k-2)!! - ... 1!!] - 1.
%C The next term is too large to include.
%C The corresponding values of k are 4, 8, 32, 180, 264, 328, 1788, 2308, 5152, 7572, 13496, ... ; all these values are even since for k odd above 11 this form is divisible by 7. - _Amiram Eldar_, Jul 18 2019
%H Amiram Eldar, <a href="/A130308/b130308.txt">Table of n, a(n) for n = 1..6</a>
%e 5 = 4!! - 3!! + 2!! - 1!! -1 = 8 - 3 + 2 - 1 - 1.
%p P:=proc(n) local a,i,j,k,w; for i from 1 by 1 to n do a:=0; for j from i by -1 to 0 do k:=j; w:=j-2; while w>0 do k:=k*w; w:=w-2; od; a:=a+k*(-1)^j od; if isprime(abs(a)-1) then print(abs(a)-1); fi; od; end: P(1000);
%t f[n_] := Sum[(-1)^(n-k)*k!!, {k, 1, n}] - 1; Select[f/@Range[32], PrimeQ] (* _Amiram Eldar_, Jul 18 2019 *)
%Y Cf. A071828, A130309.
%K nonn,bref
%O 1,1
%A _Paolo P. Lava_ and _Giorgio Balzarotti_, May 21 2007