A245757 Numbers n such that (k!+n)/k is never prime for any k.

5, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 31, 34, 37, 39, 41, 43, 47, 48, 50, 53, 54, 55, 57, 59, 61, 62, 64, 67, 69, 71, 73, 75, 76, 77, 79, 83, 86, 89, 90, 93, 94, 97, 98, 99, 101, 103, 107, 109, 110, 111, 113, 118, 119, 122, 125, 127, 128, 129, 131, 134, 137, 139, 141, 142, 143, 146

Offset: 1

Author

Derek Orr, Jul 31 2014

Keywords

• nonn

Comments

k <= n for all n so k can only be a finite set of numbers.

Only k dividing n need be considered.

By Wilson's theorem, all primes > 3 are in the sequence. - Robert Israel, Jul 31 2014

Examples

			(1!+5)/1 = 6 is not prime.
(2!+5)/2 = 7/2 is not prime.
(3!+5)/3 = 11/3 is not prime.
(4!+5)/4 = 29/4 is not prime.
(5!+5)/5 = 25 is not prime.
For any k > 5, (k!+5)/k = (k-1)! + 5/k will always be a fraction and thus, never prime. So 5 is a member of this sequence.
		

Links

• Jens Kruse Andersen, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A245756.

Programs

Maple

filter:= proc(n) local k;
  for k in numtheory:-divisors(n) do
     if isprime((k!+n)/k) then return false fi
  od:
  true
end proc:
select(filter, [$1..1000]); # Robert Israel, Jul 31 2014

Mathematica

filterQ[n_] := AllTrue[Divisors[n], !PrimeQ[(#! + n)/#]&];
Select[Range[200], filterQ] (* Jean-François Alcover, Jul 27 2020 *)

PARI

a(n)=for(k=1,n,s=(k!+n)/k;if(floor(s)==s,if(ispseudoprime(s),return(k))))
n=1;while(n<200,if(!a(n),print1(n,", "));n++)