%I #56 Aug 12 2024 13:08:50

%S 0,1,2,3,4,5,8,14,20,26,34,56,104,153,182,194,217,230,280,281,462,463,

%T 529,1445,2515,3692,6187,6851,13917,17258,48934

%N Numbers k such that k!/k# + 1 is prime, where k# is the primorial function (A034386).

%C 96835 is a term of the sequence, but its rank is not currently known. - _Serge Batalov_, Feb 06 2015

%C If k is a prime and k is a member, then k-1 is also a member, and k!/k# + 1 is the same as (k-1)!/(k-1)# + 1. See A049420. - _Jeppe Stig Nielsen_, Aug 12 2024

%H Chris Caldwell, <a href="https://t5k.org/glossary/page.php?sort=Compositorial">Compositorial</a>

%e 8!/8# + 1 = 40320/210 + 1 = 193, a prime.

%p A140294 := proc(n) local L, p, s, i; L := 1;

%p for p in select(isprime, [$2..iquo(n,2)]) do

%p s := add(i,i=convert(n,base,p)); L := L*p^((n-s)/(p-1)-1) od;

%p `if`(isprime(L+1), n, NULL) end:

%p seq(A140294(i), i=0..104); # _Peter Luschny_, Mar 27 2013

%t Primorial[p_] := Times @@ Prime[Range[PrimePi[p]]]; Select[Range[0,194], PrimeQ[#!/Primorial[#] + 1] &] (* _T. D. Noe_, Mar 27 2013 *)

%o (PARI) is(n)=ispseudoprime(n!/prod(i=1,primepi(n),prime(i))+1) \\ _Charles R Greathouse IV_, Mar 27 2013

%o (PFGW) ABC2 $a!/$a#+1

%o a: from 1 to 3000

%Y Cf. A034386, A140293, A140315, A049420, A049421, A053982.

%K nonn

%O 1,3

%A _Cino Hilliard_, May 25 2008

%E a(17)-a(25) from _Charles R Greathouse IV_, Mar 27 2013

%E a(26)-a(27) from _Giovanni Resta_, Mar 28 2013

%E a(28) from _Charles R Greathouse IV_, Mar 28 2013

%E a(29) from _Giovanni Resta_, Apr 02 2013

%E a(30) from _Roger Karpin_, Nov 29 2014

%E a(31) from _Roger Karpin_, Jun 08 2015