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

1,3

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

Table of n, a(n) for n=1..67.

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

Mathematica

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

Prog

(PARI) select( is(k)={k<3||isprime(k^2*(k-2)+k-1)}, [0..200])
(Magma) [0, 1, 2] cat  [n: n in [0..220] | IsPrime((n^2*(n-2)+n-1))]; // Vincenzo Librandi, Jul 19 2019

Crossrefs

Cf. A014206.

Sequence in context: A357425 A090034 A037016 * A239746 A101323 A298705

Adjacent sequences: A308978 A308979 A308980 * A308982 A308983 A308984

Keyword

nonn

Author

M. F. Hasler, Jul 04 2019

Status

approved