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

1,1

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

Links

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

Example

(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.

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 *)

Prog

(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++)

Crossrefs

Cf. A245756.

Sequence in context: A098761 A113837 A255363 * A240912 A263080 A280199

Adjacent sequences: A245754 A245755 A245756 * A245758 A245759 A245760

Keyword

nonn

Author

Derek Orr, Jul 31 2014

Status

approved