A075723 -id:A075723 - cp's OEIS frontend

A002384 Numbers m such that m^2 + m + 1 is prime.

1, 2, 3, 5, 6, 8, 12, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62, 66, 69, 71, 75, 77, 78, 80, 89, 90, 99, 101, 105, 110, 111, 117, 119, 131, 138, 141, 143, 147, 150, 153, 155, 161, 162, 164, 167, 168, 173, 176, 188, 189, 192, 194, 203, 206, 209, 215

Offset: 1

N. J. A. Sloane

A002383 lists the corresponding primes. - Bernard Schott, Dec 22 2012
This is also the list of bases where 111 represents a prime number. - Christian N. K. Anderson, Mar 28 2013
If d>1 divides m^2 + m + 1, then m + k*d is not in the sequence, for all k>=1. - Gionata Neri, Mar 04 2017

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 46.
L. Poletti, Le serie dei numeri primi appartenente alle due forme quadratiche (A) n^2+n+1 e (B) n^2+n-1 per l'intervallo compreso entro 121 milioni, e cioè per tutti i valori di n fino a 11000, Atti della Reale Accademia Nazionale dei Lincei, Memorie della Classe di Scienze Fisiche, Matematiche e Naturali, s. 6, v. 3 (1929), pages 193-218.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
Pantelis A. Damianou, On prime values of cyclotomic polynomials, arXiv:1101.1152 [math.NT], Jan 06 2011.
Oliver Knill, Goldbach for Gaussian, Hurwitz, Octavian and Eisenstein primes, arXiv preprint arXiv:1606.05958 [math.NT], 2016.
Oliver Knill, Some experiments in number theory, arXiv preprint arXiv:1606.05971 [math.NT], 2016.

Cf. A002383, A049407, A049408, A075723, A088503, A110284.

[ n: n in [1..300] | IsPrime(n^2+n+1)] // Vincenzo Librandi, Nov 21 2010

Select[Range@ 216, PrimeQ[#^2 + # + 1] &] (* Michael De Vlieger, Mar 06 2017 *)

is(n)=isprime(n^2+n+1) \\ Charles R Greathouse IV, Jan 21 2014

a(n) = (A088503(n) - 1)/2. - Ray Chandler

Extended by Ray Chandler, Sep 07 2005

A049407 Numbers m such that m^3 + m + 1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 12, 15, 17, 18, 21, 29, 30, 32, 39, 41, 42, 44, 48, 53, 54, 56, 60, 69, 71, 74, 77, 83, 87, 95, 102, 104, 108, 116, 117, 120, 126, 131, 135, 143, 144, 146, 152, 153, 155, 162, 168, 177, 179, 180, 186, 191, 207, 212, 219, 221, 225, 239, 240, 243

Offset: 1

N. J. A. Sloane

For s = 5, 8, 11, 14, 17, 20, ... (A016789(s) for s>=2), m_s = 1 + m + m^s is composite for m>1. Also for m=1, m_s = 3 is a prime for any s. Here we consider the case s=3.
If m == 1 (mod 3), m_s == 0 (mod 3) for any s and is not prime for m > 1. Thus for n > 1, a(n) !== 1 (mod 3) and this is true for any similar sequence based on another s value (A002384, A049408, A075723). - Jean-Christophe Hervé, Sep 20 2014
Corresponding primes are in A095692.

			3 is a term because 1 + 3 + 3^3 = 31 is a prime.

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

Cf. A002384 (s=2), A049408 (s=4), A075723 (s=6).

Cf. A095692 (corresponding primes).

[n: n in [0..300] | IsPrime(s) where s is 1+&+[n^i: i in [1..3 by 2]]]; // Vincenzo Librandi, Jun 27 2014

A049407:=n->`if`(isprime(n^3+n+1), n, NULL): seq(A049407(n), n=1..300); # Wesley Ivan Hurt, Nov 14 2014

Select[Range[500], PrimeQ[Total[#^Range[1, 3, 2]] + 1] &] (* Vincenzo Librandi, Jun 27 2014 *)

is(n)=isprime(n^3+n+1) \\ Charles R Greathouse IV, Nov 20 2012

from sympy import isprime
def ok(m): return isprime(m**3 + m + 1)
print([m for m in range(244) if ok(m)]) # Michael S. Branicky, Feb 17 2022

A049408 Numbers k such that k^4 + k + 1 is prime.

Original entry on oeis.org

1, 2, 5, 6, 9, 11, 12, 14, 24, 26, 32, 36, 44, 47, 60, 69, 72, 74, 77, 89, 90, 102, 107, 119, 126, 131, 146, 147, 159, 162, 170, 171, 186, 191, 197, 204, 206, 219, 239, 240, 252, 266, 284, 285, 290, 296, 300, 324, 347, 351, 362, 384, 426, 437, 459, 465, 470

Offset: 1

N. J. A. Sloane

For s = 5,8,11,14,17,20,..., n_s = 1 + n + n^s is always composite for any n > 1. Also for n=1, n_s=3 is a prime for any s. Here we consider the case s=4.

			26 is a term because at s=4, n=26, n_s = 1 + n + n^s = 457003 is a prime.

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

Cf. A002384, A075723, A049407.

[n: n in [0..1000] | IsPrime(s) where s is 1+n+n^4]; // Vincenzo Librandi, Jul 28 2014

Select[Range[1000], PrimeQ[1 + # + #^4] &] (* Vincenzo Librandi, Jul 28 2014 *)

for(n=1, 1000, if(isprime(1+n+n^4), print1(n",")))

A075722 Numbers n such that 1 + n + n^s is a prime, s = 7.

Original entry on oeis.org

1, 2, 15, 21, 29, 39, 42, 53, 57, 77, 81, 92, 117, 123, 131, 147, 149, 153, 167, 168, 200, 204, 207, 233, 249, 251, 252, 275, 278, 314, 317, 326, 357, 372, 378, 380, 410, 422, 434, 438, 440, 462, 467, 468, 498, 516, 546, 585, 587, 596, 608, 615, 621, 636

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=7 and find several first n's making n_s a prime (or a probable prime).

			15 is OK because at s=7, n=15, n_s = 1 + n + n^s = 170859391 is a prime.

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

Cf. A002384, A075720, A075723.

[n: n in [0..1000] | IsPrime(s) where s is 1+n+n^7]; // Vincenzo Librandi, Jul 28 2014

Select[Range[1000], PrimeQ[1 + # + #^7] &] (* Vincenzo Librandi, Jul 28 2014 *)

for(n=1,10^3,if(isprime(n^7+n+1),print1(n,", "))) \\ Derek Orr, Feb 07 2015

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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A002384 Numbers m such that m^2 + m + 1 is prime.

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A049407 Numbers m such that m^3 + m + 1 is prime.

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A049408 Numbers k such that k^4 + k + 1 is prime.

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A075722 Numbers n such that 1 + n + n^s is a prime, s = 7.

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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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