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Numbers n such that (k!+n)/k is never prime for any k.
2

%I #16 Jul 27 2020 07:35:53

%S 5,7,11,13,14,17,19,21,23,26,29,31,34,37,39,41,43,47,48,50,53,54,55,

%T 57,59,61,62,64,67,69,71,73,75,76,77,79,83,86,89,90,93,94,97,98,99,

%U 101,103,107,109,110,111,113,118,119,122,125,127,128,129,131,134,137,139,141,142,143,146

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

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

%C Only k dividing n need be considered.

%C By Wilson's theorem, all primes > 3 are in the sequence. - _Robert Israel_, Jul 31 2014

%H Jens Kruse Andersen, <a href="/A245757/b245757.txt">Table of n, a(n) for n = 1..1000</a>

%e (1!+5)/1 = 6 is not prime.

%e (2!+5)/2 = 7/2 is not prime.

%e (3!+5)/3 = 11/3 is not prime.

%e (4!+5)/4 = 29/4 is not prime.

%e (5!+5)/5 = 25 is not prime.

%e 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.

%p filter:= proc(n) local k;

%p for k in numtheory:-divisors(n) do

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

%p od:

%p true

%p end proc:

%p select(filter, [$1..1000]); # _Robert Israel_, Jul 31 2014

%t filterQ[n_] := AllTrue[Divisors[n], !PrimeQ[(#! + n)/#]&];

%t Select[Range[200], filterQ] (* _Jean-François Alcover_, Jul 27 2020 *)

%o (PARI)

%o a(n)=for(k=1,n,s=(k!+n)/k;if(floor(s)==s,if(ispseudoprime(s),return(k))))

%o n=1;while(n<200,if(!a(n),print1(n,", "));n++)

%Y Cf. A245756.

%K nonn

%O 1,1

%A _Derek Orr_, Jul 31 2014