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Largest number k > 0 such that n + k! is prime, or 0 if no such k exists.
2

%I #25 Jul 27 2020 04:44:41

%S 1,2,1,4,1,6,0,2,1,10,1,6,0,2,1,11,1,14,0,2,1,16,0,3,0,2,1,20,1,22,0,

%T 0,0,4,1,33,0,2,1,25,1,38,0,2,1,44,0,6,0,2,1,52,0,4,0,2,1,27,1,50,0,0,

%U 0,4,1,64,0,2,1,55,1,67,0,0,0,6,1,73,0,2,1,68,0,4,0,2,1,52,0,6

%N Largest number k > 0 such that n + k! is prime, or 0 if no such k exists.

%C If k >= n, then n + k! is divisible by n and is not prime.

%C a(n) < A020639(n), because if prime p divides n then p divides n + k! for k >= p. - _Robert Israel_, Aug 10 2014

%C There is no term for n = 1 since factorial primes 1 + k! can probably be arbitrarily large (A002981 shows k values). - _Jens Kruse Andersen_, Aug 13 2014

%H Jens Kruse Andersen, <a href="/A241423/b241423.txt">Table of n, a(n) for n = 2..1000</a>

%p a:= proc(n)

%p local k;

%p for k from min(numtheory:-factorset(n)) to 1 by -1 do

%p if isprime(n+k!) then return(k) fi

%p od:

%p 0

%p end proc:

%p seq(a(n),n=2..100); # _Robert Israel_, Aug 10 2014

%t a[n_] := Module[{k}, For[k = FactorInteger[n][[1, 1]], k >= 1, k--, If[PrimeQ[n + k!], Return[k]]]; 0];

%t a /@ Range[2, 100] (* _Jean-François Alcover_, Jul 27 2020, after Maple *)

%o (PARI)

%o a(n)=forstep(k=n,1,-1,if(ispseudoprime(n+k!),return(k)))

%o n=2;while(n<150,print1(a(n),", ");n++)

%Y Cf. A245714, A125162.

%K nonn

%O 2,2

%A _Derek Orr_, Aug 08 2014