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A140294 Numbers k such that k!/k# + 1 is prime, where k# is the primorial function (A034386).

%I A140294 #65 Jul 13 2025 16:29:02
%S A140294 0,1,2,3,4,5,8,14,20,26,34,56,104,153,182,194,217,230,280,281,462,463,
%T A140294 529,1445,2515,3692,6187,6851,13917,17258,48934,83515,96835
%N A140294 Numbers k such that k!/k# + 1 is prime, where k# is the primorial function (A034386).
%C A140294 96835 is a term of the sequence, but its rank is not currently known. - _Serge Batalov_, Feb 06 2015
%C A140294 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
%C A140294 All k up to 10^5 were resolved by PrimeGrid administrator "Stream" (Roman Trunov) who found a(32) and found the position of term mentioned by Batalov above (it is a(33)). - _Jeppe Stig Nielsen_, Jul 13 2025
%H A140294 Chris Caldwell, <a href="https://t5k.org/glossary/page.php?sort=Compositorial">Compositorial</a>
%e A140294 8!/8# + 1 = 40320/210 + 1 = 193, a prime.
%p A140294 A140294 := proc(n) local L, p, s, i; L := 1;
%p A140294 for p in select(isprime, [$2..iquo(n,2)]) do
%p A140294     s := add(i,i=convert(n,base,p)); L := L*p^((n-s)/(p-1)-1) od;
%p A140294 `if`(isprime(L+1), n, NULL) end:
%p A140294 seq(A140294(i), i=0..104); # _Peter Luschny_, Mar 27 2013
%t A140294 Primorial[p_] := Times @@ Prime[Range[PrimePi[p]]]; Select[Range[0,194], PrimeQ[#!/Primorial[#] + 1] &] (* _T. D. Noe_, Mar 27 2013 *)
%o A140294 (PARI) is(n)=ispseudoprime(n!/prod(i=1,primepi(n),prime(i))+1) \\ _Charles R Greathouse IV_, Mar 27 2013
%o A140294 (PFGW) ABC2 $a!/$a#+1
%o A140294 a: from 1 to 3000
%Y A140294 Cf. A034386, A140293, A140315, A049420, A049421, A053982.
%K A140294 nonn
%O A140294 1,3
%A A140294 _Cino Hilliard_, May 25 2008
%E A140294 a(17)-a(25) from _Charles R Greathouse IV_, Mar 27 2013
%E A140294 a(26)-a(27) from _Giovanni Resta_, Mar 28 2013
%E A140294 a(28) from _Charles R Greathouse IV_, Mar 28 2013
%E A140294 a(29) from _Giovanni Resta_, Apr 02 2013
%E A140294 a(30) from _Roger Karpin_, Nov 29 2014
%E A140294 a(31) from _Roger Karpin_, Jun 08 2015
%E A140294 a(32)-a(33) communicated by _Jeppe Stig Nielsen_, Jul 13 2025