A085747 Least positive k such that n! + k is a semiprime.

3, 2, 3, 1, 1, 1, 1, 1, 5, 1, 3, 17, 1, 1, 7, 2, 3, 23, 1, 1, 11, 29, 3, 1, 1, 1, 37, 1, 41, 2, 19, 11, 11, 1, 7, 3, 41, 1, 13, 127, 47, 59, 2, 37, 5, 37, 59, 1, 2, 73, 59, 79, 73, 1, 1, 61, 118, 37, 1, 61, 31, 37, 43, 7, 83, 71, 19, 7, 103, 19, 1, 107, 37, 167, 1, 103, 3, 73, 109, 97

Offset: 1

Author

Jason Earls, Jul 21 2003

Keywords

• nonn
• hard

Comments

a(106) >= 139. - Sean A. Irvine, Oct 29 2019

a(n) = 1 iff n in A078778.

Links

• Tyler Busby, Table of n, a(n) for n = 1..107 (terms 1..99, 101..103, and 105 from Max Alekseyev, terms 100 and 104 from Sean A. Irvine)
• Dario Alejandro Alpern, Factorization using the Elliptic Curve Method
• factordb, Status of 106!+139.
• Hisanori Mishima, Factorization of n! + 1 (n = 1 to 100)

Crossrefs

Cf. A089538, A089539, A089540, A089541.

Programs

Magma

m:=1; sol:=[]; for n in [1..40] do k:=1; while &+[d[2]: d in Factorization(Factorial(n)+k)] ne 2 do k:=k+1; end while; sol[m]:=k; m:=m+1; end for; sol; // Marius A. Burtea, Aug 24 2019

Mathematica

Table[SelectFirst[Range@ 1000, PrimeOmega[n! + #] == 2 &], {n, 40}] (* Michael De Vlieger, Mar 08 2016, Version 10 *)
lpk[n_]:=Module[{k=1,fn=n!},While[PrimeOmega[fn+k]!=2,k++];k]; Array[lpk,80] (* Harvey P. Dale, Feb 18 2025 *)

PARI

a(n) = {k = 1; while (bigomega(n!+k) != 2, k++); k;} \\ Michel Marcus, Mar 08 2016

Extensions

Extended by Robert G. Wilson v, Jul 27 2003

a(55) from Ray Chandler, Nov 09 2003

a(56)-a(80) from Sean A. Irvine, Mar 29 2010