%I #42 May 21 2021 16:41:09
%S 0,3,4,5,7,10,12,20,37,52,73,149,304,540,2135,7112,7436,9357
%N Numbers n such that !n + 7 is prime.
%C At present the terms >= 2135 are only probable primes.
%C Expected to be finite, similar to Živković (1999).
%H M. Živković, <a href="https://doi.org/10.1090/S0025-5718-99-00990-4">The number of primes Sum_{i=1..n} (-1)^(n-i)*i! is finite</a>, Math. Comp. 68 (1999), no. 225, 403-409.
%e 4 is a term, because 0! + 1! + 2! + 3! + 7 = 17 is prime.
%t Do[ If[ PrimeQ[ Sum[ k!, {k, 0, n - 1} ] + 7 ], Print[ n ] ], {n, 1, 600} ]
%t Join[{0},Flatten[Position[Accumulate[Range[0,600]!]+7,_?PrimeQ]]] (* The program generates the first 14 terms. To generate more increase the Range constant, but the program may take a long time to run. *) (* _Harvey P. Dale_, May 21 2021 *)
%o (PARI) s=0; for(n=0,600,if(ispseudoprime(s + 7),print1(n,", ")); s+=n!)
%Y Cf. A001272, A063833, A100614.
%K nonn,hard,more
%O 1,2
%A _Serge Batalov_, Jul 14 2017