A308981 Nonnegative integers k such that k^3 - 2*k^2 + k - 1 is not composite.

0, 1, 2, 3, 5, 6, 7, 10, 12, 13, 15, 20, 23, 26, 27, 28, 30, 33, 35, 37, 38, 41, 45, 48, 50, 56, 61, 63, 65, 66, 70, 71, 72, 82, 83, 85, 90, 96, 98, 107, 108, 115, 120, 122, 126, 128, 133, 140, 141, 142, 145, 148, 156, 160, 162, 166, 173, 175, 180, 185, 191, 202, 205, 208, 213, 217, 220

Offset: 1

Author

M. F. Hasler, Jul 04 2019

Keywords

• nonn

Comments

Apart the three initial terms which lead to +/-1, all other terms lead to prime P(k) = k^3 - 2*k^2 + k - 1.

The polynomial Q = (((x^2-k)^2-k)^2-x-k)/(x^2 - x - k) of degree 6 has two factors of degree <= 3 when k is in A014206. This can only happen when the constant term of Q, which equals -P(k), is not prime. Therefore, A014206 is a subsequence of the complement of this sequence.

Links

• R. Israel, in reply to T. Baruchel, A014206 and computer algebra systems, SeqFan list, July 4, 2019.

Crossrefs

Cf. A014206.

Programs

Magma

[0,1,2] cat  [n: n in [0..220] | IsPrime((n^2*(n-2)+n-1))]; // Vincenzo Librandi, Jul 19 2019

Mathematica

Join[{0, 1, 2}, Select[Range[230], PrimeQ[((#^2 (# - 2) + # - 1))] &]] (* Vincenzo Librandi, Jul 19 2019 *)

PARI

select( is(k)={k<3||isprime(k^2*(k-2)+k-1)}, [0..200])