A075718 -id:A075718 - cp's OEIS frontend

A075717 1+n+n^13 is a prime.

1, 5, 15, 39, 50, 56, 105, 116, 128, 153, 168, 170, 243, 245, 264, 308, 314, 369, 401, 429, 480, 489, 531, 551, 599, 608, 680, 690, 699, 701, 785, 804, 939, 978, 1050, 1056, 1065, 1073, 1110, 1169, 1224, 1226, 1271, 1283, 1308, 1310, 1391, 1401

Offset: 1

Zak Seidov, Oct 03 2002

For s = 5,8,11,14,17,20,..., n_s=1+n+n^s is always composite for any n>1. Also at n=1, n_s=3 is a prime for any s. So it is interesting to consider only the cases of s =/= 5,8,11,14,17,20,... and n>1. Here I consider the case s=13 and find several first n's making n_s a prime (or a probable prime).

			5 is OK because at s=13, n=2, n_s=1+n+n^s=1220703131 is a prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1900

Cf. A002384, A075716, A075718.

[n: n in [0..2000] | IsPrime(s) where s is 1+n+n^13]; // Vincenzo Librandi, Jul 28 2014

Select[Range[1500], PrimeQ[1 + # + #^13] &] (* Harvey P. Dale, Apr 20 2013 *)

n=0; for(k=1, 60, n=n+1; while(!isprime(1+n+n^13), n=n+1); print1(n","))

More terms from Ralf Stephan, Mar 20 2003

A075719 1+n+n^s is a prime, s=10.

Original entry on oeis.org

1, 3, 8, 21, 23, 26, 33, 36, 38, 45, 51, 57, 69, 71, 78, 92, 107, 117, 149, 156, 170, 176, 179, 195, 209, 216, 219, 224, 261, 293, 321, 341, 359, 374, 378, 386, 390, 404, 410, 413, 420, 474, 492, 507, 516, 546, 569, 572, 582, 621, 632, 683, 767, 783, 789, 809

Offset: 1

Zak Seidov, Oct 03 2002

For s = 5,8,11,14,17,20,..., n_s=1+n+n^s is always composite for any n>1. Also at n=1, n_s=3 is a prime for any s. So it is interesting to consider only the cases of s =/= 5,8,11,14,17,20,... and n>1. Here i consider the case s=10 and find several first n's making n_s a prime (or a probable prime).

			3 is OK because at s=10, n=3, n_s=1+n+n^s=59053 is a prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..3300

Cf. A002384, A075718, A075720.

[n: n in [0..1000] | IsPrime(s) where s is 1+n+n^10]; // Vincenzo Librandi, Jul 28 2014

Select[Range[1000], PrimeQ[1 + # + #^10] &] (* Vincenzo Librandi, Jul 28 2014 *)

for(n=1,1000,if(isprime(1+n+n^10),print1(n",")))

More terms from Ralf Stephan, Apr 05 2003

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

Original entry on oeis.org

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

Derek Orr, Jul 23 2014

nonn

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

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

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

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).

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

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

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A075717 1+n+n^13 is a prime.

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A075719 1+n+n^s is a prime, s=10.

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Magma

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

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