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A245476 Least number k > 1 such that k^n + k + 1 is prime, or 0 if no such number exists. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 7 12:56 EDT 2021. Contains 343650 sequences. (Running on oeis4.)