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A140293 Numbers k such that k!/k#-1 is prime, where k# is the primorial function (A034386).

%I A140293 #39 Aug 12 2024 13:08:41
%S A140293 4,5,6,7,8,16,17,21,34,39,45,50,72,73,76,133,164,202,216,221,280,281,
%T A140293 496,605,2532,2967,3337,8711,10977,13724,15250,18160,20943,33684,41400
%N A140293 Numbers k such that k!/k#-1 is prime, where k# is the primorial function (A034386).
%C A140293 a(31) > 14000. - _Giovanni Resta_, Apr 02 2013
%C A140293 a(36) > 50000. - _Roger Karpin_, Jul 07 2015
%C A140293 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 A049421. - _Jeppe Stig Nielsen_, Aug 12 2024
%H A140293 Chris Caldwell, <a href="https://t5k.org/glossary/page.php?sort=Compositorial">Compositorial</a>
%H A140293 Chris Caldwell, <a href="https://t5k.org/primes/search.php?Comment=Compositorial&amp;Style=HTML&amp;OnList=all">Compositorial list search</a>
%F A140293 n such that n!/n# - 1 is prime, where n# is the primorial function n# = product(i = 1 .. pi(n), prime(i)), where pi(n) is the prime counting function.
%e A140293 7!/7# = 5040/210 = 24. 24 - 1 = 23, which is prime.
%t A140293 Select[Range[16], PrimeQ[#!/(Times@@Prime[Range[PrimePi[#]]]) - 1] &] (* _Alonso del Arte_, Nov 28 2014 *)
%o A140293 (PARI) g(n) = for(x=4,n,y=x!/primorial(x)-1;z=nextprime(y+1); if(ispseudoprime(y),print1(x",")))
%Y A140293 Cf. A140294, A140315, A049420, A049421, A049614, A057017.
%K A140293 nonn,more
%O A140293 1,1
%A A140293 _Cino Hilliard_, May 25 2008
%E A140293 a(18)-a(27) from _Giovanni Resta_, Mar 28 2013
%E A140293 a(28)-a(30) from _Giovanni Resta_, Apr 02 2013
%E A140293 a(31) from _Roger Karpin_, Nov 28 2014
%E A140293 a(32)-a(33) from Daniel Heuer, ca Aug 2000
%E A140293 a(34)-a(35) from _Serge Batalov_, Feb 09 2015