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

4, 5, 6, 7, 8, 16, 17, 21, 34, 39, 45, 50, 72, 73, 76, 133, 164, 202, 216, 221, 280, 281, 496, 605, 2532, 2967, 3337, 8711, 10977, 13724, 15250, 18160, 20943, 33684, 41400

Offset: 1

Author

Cino Hilliard, May 25 2008

Keywords

• nonn
• more

Comments

a(31) > 14000. - Giovanni Resta, Apr 02 2013

a(36) > 50000. - Roger Karpin, Jul 07 2015

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

Examples

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

Links

• Chris Caldwell, Compositorial
• Chris Caldwell, Compositorial list search

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

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.

Extensions

a(18)-a(27) from Giovanni Resta, Mar 28 2013

a(28)-a(30) from Giovanni Resta, Apr 02 2013

a(31) from Roger Karpin, Nov 28 2014

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

a(34)-a(35) from Serge Batalov, Feb 09 2015