A078778 Numbers n such that n!+1 is a semiprime.

4, 5, 6, 7, 8, 10, 13, 14, 19, 20, 24, 25, 26, 28, 34, 38, 48, 54, 55, 59, 71, 75, 92, 109, 114, 115

Offset: 1

Author

Jason Earls, Jan 09 2003

Keywords

• more
• nonn

Comments

Subsequence of (and likely is equal to) the union of A146968 and A181764. - Max Alekseyev, May 28 2015

Note that the two prime factors of 38!+1 = 523022617466601111760007224100074291200000001 = 14029308060317546154181 * 37280713718589679646221 both have 23 decimal digits. Are there any other terms in this sequence other than 4,5,7 and 38 with this property?

a(27) > 139. - Robert Price, Apr 11 2019

Other terms in this sequence: 392, 551, 601, 770, 772, 878, 1033, 1320, 1831, 2620, 2808, 3752, 4233, 4616, 4984, 7260. - Chai Wah Wu, Feb 28 2020

Examples

			4 is in the sequence because 4!+1=25=5*5 is semiprime. But 9 is not in the sequence because 9!+1=19*71*269 is not semiprime. - _Sean A. Irvine_, Nov 15 2009
		

Links

• FactorDB, Status of 140!+1.

Crossrefs

Cf. A001358, A082952, A090159, A090160, A078781.

Programs

Magma

IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..60] | IsSemiprime(Factorial(n)+1)]; // Vincenzo Librandi, May 26 2015

Mathematica

Select[Range[100], Plus@@Last/@FactorInteger[#! + 1]==2 &] (* Vincenzo Librandi, May 26 2015 *)
Select[Range[100],PrimeOmega[#!+1]==2&] (* Harvey P. Dale, Mar 19 2017 *)

PARI

{ fp(a,b)=local(c,d,r); for(n=a,b,r=n!+1; c=vecmin(factor(r)[, 1]~); d=vecmax(factor(r)[,1]~); if(bigomega(r)==2 && isprime(c) && isprime(d), print1(n" ");)) } fp(1,100)

Extensions

Term 109 from Sean A. Irvine, Nov 15 2009

Term 114 (factored by Womack et al.) from Sean A. Irvine, May 25 2015

Term 115 (factored by Womack et al.) from Sean A. Irvine, Feb 08 2016