A245476 Least number k > 1 such that k^n + k + 1 is prime, or 0 if no such number exists.

2, 2, 2, 2, 0, 2, 2, 0, 3, 3, 0, 2, 5, 0, 2, 2, 0, 2, 8, 0, 6, 3, 0, 6, 15, 0, 6, 2, 0, 2, 23, 0, 23, 56, 0, 15, 114, 0, 14, 11, 0, 3, 14, 0, 29, 110, 0, 21, 9, 0, 53, 59, 0, 6, 2, 0, 3, 29, 0, 71, 21, 0, 146, 17, 0, 35, 2, 0, 9, 6, 0, 77, 41, 0, 27, 176, 0, 153, 21, 0, 39, 32, 0, 2, 314, 0, 3, 5, 0, 66, 44, 0, 234

Offset

1,1

Comments

Except for a(2), a(n) = 0 if n == 2 mod 3 (A016789).

It appears that this is an "if and only if".

a(n) = 2 if and only if n is in A057732.

Many terms in the linked table correspond to probable primes. If n == 2 mod 3 then k^2+k+1 divides k^n+k+1. This is why a(n) = 0 if n > 2 and n == 2 mod 3. - Jens Kruse Andersen, Jul 28 2014

Links

Robert Israel and Jens Kruse Andersen, Table of n, a(n) for n = 1..1000 (first 640 terms from Robert Israel)

Example

2^9 + 2 + 1 = 515 is not prime. 3^9 + 3 + 1 = 19687 is prime. Thus a(9) = 3.

Maple

f:= proc(n) local k;
   if n mod 3 = 2 and n > 2 then return 0 fi;
   for k from 2 to 10^6 do
      if isprime(k^n+k+1) then return k fi
   od:
  error("no solution found for n = %1", n);
end proc:
seq(f(n), n=1..100); # Robert Israel, Jul 27 2014

Prog

(PARI) a(n) = if(n>2&&n==Mod(2, 3), return(0)); k=2; while(!ispseudoprime(k^n+k+1), k++); k
vector(150, n, a(n)) \\ Derek Orr with corrections and improvements from Colin Barker, Jul 23 2014

Crossrefs

Cf. A127599, A016789, A057732.

Cf. Numbers n such that n^s + n + 1 is prime: A005097 (s = 1), A002384 (s = 2), A049407 (s = 3), A049408 (s = 4), A075723 (s = 6), A075722 (s = 7), A075720 (s = 9), A075719 (s = 10), A075718 (s = 12), A075717 (s = 13), A075716 (s = 15), A075715 (s = 16), A075714 (s = 18), A075713 (s = 19).

Sequence in context: A205777 A053398 A065833 * A215884 A305029 A097033

Adjacent sequences: A245473 A245474 A245475 * A245477 A245478 A245479

Keyword

nonn

Author

Derek Orr, Jul 23 2014

Status

approved