A140293 - OEIS

%I #39 Aug 12 2024 13:08:41

%S 4,5,6,7,8,16,17,21,34,39,45,50,72,73,76,133,164,202,216,221,280,281,

%T 496,605,2532,2967,3337,8711,10977,13724,15250,18160,20943,33684,41400

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

%C a(31) > 14000. - _Giovanni Resta_, Apr 02 2013

%C a(36) > 50000. - _Roger Karpin_, Jul 07 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 A049421. - _Jeppe Stig Nielsen_, Aug 12 2024

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

%H Chris Caldwell, <a href="https://t5k.org/primes/search.php?Comment=Compositorial&amp;Style=HTML&amp;OnList=all">Compositorial list search</a>

%F 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 7!/7# = 5040/210 = 24. 24 - 1 = 23, which is prime.

%t Select[Range[16], PrimeQ[#!/(Times@@Prime[Range[PrimePi[#]]]) - 1] &] (* _Alonso del Arte_, Nov 28 2014 *)

%o (PARI) g(n) = for(x=4,n,y=x!/primorial(x)-1;z=nextprime(y+1); if(ispseudoprime(y),print1(x",")))

%Y Cf. A140294, A140315, A049420, A049421, A049614, A057017.

%K nonn,more

%O 1,1

%A _Cino Hilliard_, May 25 2008

%E a(18)-a(27) from _Giovanni Resta_, Mar 28 2013

%E a(28)-a(30) from _Giovanni Resta_, Apr 02 2013

%E a(31) from _Roger Karpin_, Nov 28 2014

%E a(32)-a(33) from Daniel Heuer, ca Aug 2000

%E a(34)-a(35) from _Serge Batalov_, Feb 09 2015