1,1

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n) = m*x(n-1) + x(n-2) for k >= 2. For primes p, we have (a) p divides x(p-((m^2+4)/p)); (b) x(p) == ((m^2+4)/p) (mod p), where (D/p) is the Kronecker symbol. This sequence gives composite numbers k such that gcd(k, m^2+4) = 1 and that conditions similar to (a) and (b) hold for k simultaneously, where m = 2.

If k is not required to be coprime to m^2 + 4 (= 8), then there are 1190 such k <= 10^5 and 4847 such k <= 10^6, while there are only 41 terms <= 10^5 and 119 terms <= 10^6 in this sequence.

From Amiram Eldar, Jan 23 2026: (Start)

Called "Pell-Pell-Lucas pseudoprimes" by Andrica and Bagdasar (2020, 2021, 2022).

They conjectured that this sequence is infinite, and their conjecture was proved by Grantham (2021).

These are the terms k in A330276 ("Pell-Lucas pseudoprimes") such that k | P(k - (-1)^((k^2-1)/8)), where P = A000129 is the sequence of Pell numbers. (End)

Daniel Suteu, Table of n, a(n) for n = 1..10000

Dorin Andrica and Ovidiu Bagdasar, Arithmetic and Trigonometric Properties of Some Classical Recurrent Sequences, in: Recurrent Sequences: Key Results, Applications, and Problems, Springer, Cham, 2020. See Chapter 3, Remark 3.7, p. 94.

Dorin Andrica and Ovidiu Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterranean Journal of Mathematics, Vol. 18, No. 2 (2021), pp. 47. See Definition 5.2, p. 47.

Dorin Andrica and Ovidiu Bagdasar, Pseudoprimality related to the generalized Lucas sequences, Mathematics and Computers in Simulation, Vol. 201 (2022), pp. 528-542; alternative link. See Section 3.2.5.

Dorin Andrica, Ovidiu Bagdasar, and Michael Th. Rassias, Weak Pseudoprimality Associated with the Generalized Lucas Sequences, in: N. J. Daras and T. M. Rassias (eds.), Approximation and Computation in Science and Engineering, Springer Optimization and Its Applications, Vol. 180, Springer, Cham, 2022, pp. 53-75. See Section 2.3, p. 61.

Jon Grantham, Proof of Two Conjectures of Andrica and Bagdasar, Integers, Vol. 21 (2021), Article #A111.

169 divides Pell(168) as well as Pell(169) - 1, so 169 is a term.

(PARI) pellmod(n, m)=((Mod([2, 1; 1, 0], m))^n)[1, 2]

isA327652(n)=!isprime(n) && pellmod(n, n)==kronecker(8, n) && !pellmod(n-kronecker(8, n), n) && gcd(n, 8)==1 && n>1

m m=1 m=2 m=3

k | x(k-Kronecker(m^2+4,k))* A081264 U A141137 A327651 A327653

k | x(k)-Kronecker(m^2+4,k) A049062 A099011 A327654

both A212424 this seq. A327655

* k is composite and coprime to m^2 + 4.

Cf. A000129, A091337 ({Kronecker(8,n)}).

Subsequence of A330276.

Sequence in context: A099011 A330276 A351337 * A112076 A305055 A069645

Adjacent sequences: A327649 A327650 A327651 * A327653 A327654 A327655

Jianing Song, Sep 20 2019

approved