A078781 Numbers n such that n!-1 is a semiprime.

5, 8, 10, 13, 16, 20, 23, 24, 26, 27, 34, 36, 40, 47, 50, 59, 68, 79, 85, 93, 137, 143, 151

Offset: 1

Author

Jason Earls, Jan 09 2003

Keywords

• nonn
• more
• hard

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.

Crossrefs

Cf. A001358, A080802.

Cf. A078778 (numbers such that n!+1 is a semiprime).

Cf. also A007623, A070939, A266344.

Programs

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

Mathematica

Select[Range[50], PrimeOmega[#! - 1] == 2 &] (* Vincenzo Librandi, Dec 28 2015 *)

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)

Extensions

More terms from Hugo Pfoertner, Apr 05 2003

a(23) added by Daniel Suteu, Mar 30 2019