A153439 Numbers k such that k^6 + k^3 + 1 is prime.

1, 2, 3, 8, 11, 20, 21, 26, 30, 50, 51, 56, 60, 78, 98, 102, 117, 129, 134, 146, 159, 171, 186, 189, 191, 198, 200, 209, 210, 212, 222, 240, 249, 267, 269, 278, 279, 299, 300, 333, 344, 363, 383, 390, 398, 399, 425, 429, 438, 444, 450, 458, 504, 509, 533, 540

Offset: 1

Author

Pierre CAMI, Dec 26 2008

Keywords

• nonn

Comments

n^6+n^3+1 is the cyclotomic polynomial Phi(9,n). Damianou, p. 4, claims that there is strong experimental evidence that k^6 + k^3 + 1 is prime for infinitely many values of k. - Jonathan Vos Post, Jan 06 2011 [Comment corrected by Jon E. Schoenfield, Jan 22 2018]

Links

• Pierre CAMI, Table of n, a(n) for n = 1..37835
• Pantelis A. Damianou, On prime values of cyclotomic polynomials, arXiv:1101.1152 [math.NT], 2011.

Crossrefs

Cf. A153438, A164989.

Programs

Magma

[n: n in [0..500] | IsPrime(n^3*(n^3+1)+1)] // Vincenzo Librandi, Nov 26 2010

Mathematica

Select[Range@ 600, PrimeQ[#^6 + #^3 + 1] &] (* Michael De Vlieger, Jan 22 2018 *)

PARI

is(n)=prime(n^6+n^3+1) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

Some unclear comments deleted by N. J. A. Sloane, Sep 06 2009

More terms from Vincenzo Librandi, Mar 25 2010