A078778 - OEIS

%I #39 Sep 08 2022 08:45:08

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

%T 114,115

%N Numbers n such that n!+1 is a semiprime.

%C Subsequence of (and likely is equal to) the union of A146968 and A181764. - _Max Alekseyev_, May 28 2015

%C 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?

%C a(27) > 139. - _Robert Price_, Apr 11 2019

%C 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

%H FactorDB, <a href="http://factordb.com/index.php?query=140%21%2B1">Status of 140!+1</a>.

%e 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

%t Select[Range[100], Plus@@Last/@FactorInteger[#! + 1]==2 &] (* _Vincenzo Librandi_, May 26 2015 *)

%t Select[Range[100],PrimeOmega[#!+1]==2&] (* _Harvey P. Dale_, Mar 19 2017 *)

%o (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)

%o (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

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

%K more,nonn

%O 1,1

%A _Jason Earls_, Jan 09 2003

%E Term 109 from _Sean A. Irvine_, Nov 15 2009

%E Term 114 (factored by Womack et al.) from _Sean A. Irvine_, May 25 2015

%E Term 115 (factored by Womack et al.) from _Sean A. Irvine_, Feb 08 2016