A245756 -id:A245756 - cp's OEIS frontend

A240163 Numbers n such that (k!+n)/k is prime for some k.

1, 2, 3, 4, 6, 8, 9, 10, 12, 15, 16, 18, 20, 22, 24, 25, 27, 28, 30, 32, 33, 35, 36, 38, 40, 42, 44, 45, 46, 49, 51, 52, 56, 58, 60, 63, 65, 66, 68, 70, 72, 74, 78, 80, 81, 82, 84, 85, 87, 88, 91, 92, 95, 96, 100, 102, 104, 105, 106, 108, 112, 114, 115, 116, 117, 120, 121, 123, 124, 126

Offset: 1

Derek Orr, Aug 01 2014

nonn

Complement of A245757.

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

Cf. A245757, A245756.

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++)

Typo in PARI code fixed by Colin Barker, Aug 02 2014

A240160 Least number k such that (k!-n)/k is prime, or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 4, 5, 0, 7, 8, 0, 0, 0, 4, 13, 0, 15, 4, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 31, 0, 33, 34, 5, 0, 0, 0, 39, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 6, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5

Offset: 1

Derek Orr, Aug 01 2014

nonn

For a(n) > 0, a(n) is a divisor of n.

If a(n) = n, then n - 1 is in A002982.

Jens Kruse Andersen, Table of n, a(n) for n = 1..2500

Cf. A002982, A240161, A245756.

a(n)=for(k=1,n,s=(k!-n)/k;if(floor(s)==s,if(ispseudoprime(s),return(k))))
vector(150, n, a(n))

Typo in PARI code fixed by Colin Barker, Aug 02 2014

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

Original entry on oeis.org

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

Derek Orr, Jul 31 2014

nonn

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

			(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.

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

Cf. A245756.

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

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

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++)

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A240163 Numbers n such that (k!+n)/k is prime for some k.

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A240160 Least number k such that (k!-n)/k is prime, or 0 if no such k exists.

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

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