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

%I A085747 #55 Mar 08 2025 09:54:52
%S A085747 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,
%T A085747 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,
%U A085747 118,37,1,61,31,37,43,7,83,71,19,7,103,19,1,107,37,167,1,103,3,73,109,97
%N A085747 Least positive k such that n! + k is a semiprime.
%C A085747 a(106) >= 139. - _Sean A. Irvine_, Oct 29 2019
%C A085747 a(n) = 1 iff n in A078778.
%H A085747 Tyler Busby, <a href="/A085747/b085747.txt">Table of n, a(n) for n = 1..107</a> (terms 1..99, 101..103, and 105 from Max Alekseyev, terms 100 and 104 from Sean A. Irvine)
%H A085747 Dario Alejandro Alpern, <a href="https://www.alpertron.com.ar/ECM.HTM">Factorization using the Elliptic Curve Method</a>
%H A085747 factordb, <a href="http://factordb.com/index.php?query=106%21%2B139">Status of 106!+139</a>.
%H A085747 Hisanori Mishima, <a href="https://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha104.htm">Factorization of n! + 1 (n = 1 to 100)</a>
%t A085747 Table[SelectFirst[Range@ 1000, PrimeOmega[n! + #] == 2 &], {n, 40}] (* _Michael De Vlieger_, Mar 08 2016, Version 10 *)
%t A085747 lpk[n_]:=Module[{k=1,fn=n!},While[PrimeOmega[fn+k]!=2,k++];k]; Array[lpk,80] (* _Harvey P. Dale_, Feb 18 2025 *)
%o A085747 (PARI) a(n) = {k = 1; while (bigomega(n!+k) != 2, k++); k;} \\ _Michel Marcus_, Mar 08 2016
%o A085747 (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
%Y A085747 Cf. A089538, A089539, A089540, A089541.
%K A085747 nonn,hard
%O A085747 1,1
%A A085747 _Jason Earls_, Jul 21 2003
%E A085747 Extended by _Robert G. Wilson v_, Jul 27 2003
%E A085747 a(55) from _Ray Chandler_, Nov 09 2003
%E A085747 a(56)-a(80) from _Sean A. Irvine_, Mar 29 2010