OFFSET
1,1
COMMENTS
The next candidate for a continuation is 154!-1, which is composite with 272 decimal digits and unknown factorization. Further known terms are 157, 229, 381, 390, 392, 400, 814, 929; factorization unknown for 154, 196, 232, 271, 307, 322, 332, 333, 334, 350, 352, 386, 389, 443, 449, ...
Note that the two prime factors of 24!-1 = 620448401733239439359999 = 625793187653 * 991459181683 both have 12 decimal digits.
There is another term with prime factors with equal number of decimal digits: 34!-1 = 10398560889846739639*28391697867333973241 (20 digits each)
From Antti Karttunen, Dec 27 2015: (Start)
Furthermore, both factors of 24!-1 are in binary system 40 bits long (A070939), and in factorial base representation (A007623) they both have 14 digits: <7,2,6,5,4,8,2,3,0,0,2,0,2,1> and <11,5,2,10,1,5,6,3,4,1,1,3,0,1>. That is, A007623(625793187653) = 72654823002021, but the latter number cannot be represented reliably in such a more compact form, because it already contains digits > 9.
Factors of 34!-1 are 64 and 65 bits long, and their factorial base representations contain both 20 digits: <4,5,9,3,1,13,11,7,9,1,0,6,1,1,6,5,3,1,0,1> and <11,13,7,10,0,12,3,4,6,11,1,8,1,4,2,2,1,2,2,1>.
Also the factors of 5!-1 = 119 = 7*17 are both of the same length in factorial base system: "101" and "221".
(End)
1338, 1447, 1788, 1824, 2805, 2881, 2960, 5824 are also terms of the sequence. - Chai Wah Wu, Feb 28 2020
LINKS
FactorDB, Status of 154!-1.
Paul Leyland, Tables of factors of N!+1 and N!-1.
Andrew Walker, Factors of n!-1 for n>=400.
MATHEMATICA
Select[Range[50], PrimeOmega[#! - 1] == 2 &] (* Vincenzo Librandi, Dec 28 2015 *)
PROG
(PARI) { fm(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(2, 100)
(Magma) IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [3..50] | IsSemiprime(Factorial(n)-1)]; // Vincenzo Librandi, Dec 28 2015
CROSSREFS
KEYWORD
nonn,more,hard
AUTHOR
Jason Earls, Jan 09 2003
EXTENSIONS
More terms from Hugo Pfoertner, Apr 05 2003
a(23) added by Daniel Suteu, Mar 30 2019
STATUS
approved