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

1,1

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

Links

Table of n, a(n) for n=1..17.

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.

Example

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

Mathematica

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

Crossrefs

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

Sequence in context: A274556 A066235 A037365 * A118733 A328118 A047275

Adjacent sequences: A083337 A083338 A083339 * A083341 A083342 A083343

Keyword

nonn,more,hard

Author

Hugo Pfoertner, Apr 24 2003

Extensions

a(15) from Hugo Pfoertner, Jul 13 2019

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

Status

approved