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%I #17 Jul 01 2020 19:54:19
%S 3,4,7,21,60
%N Numbers k such that prime(k)!/prime(k)# - 1 is prime.
%C No more terms through 1000. - _Ryan Propper_, Jan 27 2007
%e 1*2*3*4*5/(2*3*5) - 1 = 3, a prime, so a(1)=3 as 5=prime(3);
%e 1*2*3*4*5*6*7/(2*3*5*7) - 1 = 23, a prime, so a(2)=4 as 7=prime(4);
%e 1*2*3*4*5*6*7*8*9*10*11*12*13*14*15*16*17/(2*3*5*7*11*13*17) - 1 = 696729599, a prime, so a(3)=7 as 17=prime(7).
%t Primorial[n_] := Times @@ Select[Range[n], PrimeQ]; Do[k = Prime[n]; If[PrimeQ[k!/Primorial[k] - 1], Print[n]], {n, 10^3}] (* _Ryan Propper_, Jan 27 2007 *)
%t With[{nn=70},Position[#[[1]]/#[[2]]-1&/@Thread[{Prime[Range[ nn]]!,FoldList[ Times,Prime[Range[nn]]]}],_?PrimeQ]//Flatten] (* _Harvey P. Dale_, Jul 01 2020 *)
%o (PARI) isok(k) = isprime(prime(k)!/prod(i=1, k, prime(i)) - 1); \\ _Michel Marcus_, Sep 15 2019
%Y Cf. A092435 (p!/p#), A124083.
%K more,nonn
%O 1,1
%A _Pierre CAMI_, Nov 25 2006
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