A075719 -id:A075719 - cp's OEIS frontend

A075718 1+n+n^s is a prime, s=12.

1, 2, 14, 30, 32, 44, 65, 87, 90, 122, 134, 149, 162, 186, 189, 227, 237, 249, 255, 266, 311, 354, 366, 456, 476, 485, 561, 567, 584, 597, 605, 650, 665, 672, 689, 720, 771, 819, 884, 899, 975, 990, 1059, 1082, 1092, 1191, 1200, 1241, 1257, 1295, 1347, 1367

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=12 and find several first n's making n_s a prime (or a probable prime).

			2 is OK because at s=12, n=2, n_s=1+n+n^s=4099 is a prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..2200

Cf. A002384, A075717, A075719.

[n: n in [0..2000] | IsPrime(s) where s is 1+n+n^12]; // Vincenzo Librandi, Jul 28 2014

Select[Range[2000], PrimeQ[1 + # + #^12] &] (* Vincenzo Librandi, Jul 28 2014 *)

for(n=1,1000,if(isprime(1+n+n^12),print1(n",")))

More terms from Ralf Stephan, Apr 05 2003

A075720 Numbers n such that n^9 + n + 1 is a prime.

Original entry on oeis.org

1, 3, 9, 11, 14, 15, 18, 23, 38, 51, 66, 89, 95, 140, 170, 185, 186, 194, 239, 258, 294, 315, 345, 366, 384, 386, 393, 401, 404, 408, 429, 459, 485, 495, 506, 531, 573, 611, 614, 665, 675, 678, 680, 683, 695, 750, 771, 791, 849, 870, 879, 941, 954, 1016, 1086

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=9 and find several first n's making n_s a prime (or a probable prime).

			9 is a member because at n=9, 1 + n + n^s = 387420499 is a prime.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A002384, A075719, A075722.

[n: n in [0..1500] | IsPrime(s) where s is 1+n+n^9]; // Vincenzo Librandi, Jul 28 2014

Select[Range[1500], PrimeQ[1 + # + #^9] &] (* Vincenzo Librandi, Jul 28 2014 *)

for(n=1,10^3,if(isprime(n^9+n+1),print1(n,", "))) \\ Derek Orr, Feb 07 2015

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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A075718 1+n+n^s is a prime, s=12.

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A075720 Numbers n such that n^9 + n + 1 is a prime.

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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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