A083340 Numbers n such that A020549(n)=(n!)^2+1 is a semiprime.

6, 7, 8, 12, 15, 16, 17, 18, 19, 28, 29, 41, 45, 53, 55, 61, 73

Offset: 1

Author

Hugo Pfoertner, Apr 24 2003

Keywords

• nonn
• more
• hard

Comments

The smaller of the two prime factors is given in A083341. The next candidates for a continuation are 55 and 61. (55!)^2 + 1 is composite with 147 decimal digits and unknown factorization.

(55!)^2 + 1 has been factored using ECM into P52*P96 with P52 = A083341(15). (61!)^2 + 1 is composite with 168 decimal digits. - Hugo Pfoertner, Jul 13 2019

Using CADO-NFS, (61!)^2 + 1 has been factored into P58*P110 with P58 = A282706(61) in 17 days wall clock time using 56 million CPU seconds. a(18) >= 75. - Hugo Pfoertner, Aug 04 2019

Examples

			a(1)=6 because (6!)^2+1=518401=13*39877 is a semiprime.
		

Links

• The CADO-NFS Development Team, Cado-NFS, An Implementation of the Number Field Sieve Algorithm, Release 2.3.0, 2017
• Andrew Walker, Factors of (n!)^2+1.
• factordb.com, Status of 75!^2+1.

Crossrefs

Cf. A020549, A001358, A083341, A046029, A282706.

Programs

Mathematica

Select[Range[60],PrimeOmega[(#!)^2+1]==2&] (* Harvey P. Dale, Dec 12 2018 *)

Extensions

a(15) from Hugo Pfoertner, Jul 13 2019

a(16), a(17) from Hugo Pfoertner, Aug 04 2019