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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

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.

Examples

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

Links

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

Crossrefs

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

Cf. A095692 (corresponding primes).

Programs

Magma

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

Maple

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

Mathematica

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

PARI

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

Python

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