A339521 Odd composite integers m such that A005248(2*m-J(m,5)) == 3 (mod m), where J(m,5) is the Jacobi symbol.

21, 203, 323, 329, 377, 451, 609, 861, 1001, 1081, 1183, 1547, 1729, 1819, 1891, 2033, 2211, 2821, 3081, 3549, 3653, 3827, 4089, 4181, 4669, 5671, 5777, 5887, 6601, 6721, 8149, 8557, 8841, 10877, 11309, 11663, 13201, 13601, 13861, 13981, 14701, 15051, 15251

Offset

1,1

Comments

The generalized Pell-Lucas sequences of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy V(k*p-J(p,D)) == V(k-1) (mod p) whenever p is prime, k is a positive integer, b=1 and D=a^2-4.

The composite integers m with the property V(k*m-J(m,D)) == V(k-1) (mod m) are called generalized Pell-Lucas pseudoprimes of level k+ and parameter a.

Here b=1, a=3, D=5 and k=2, while V(m) recovers A005248(m).

References

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.

D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).

D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Links

Table of n, a(n) for n=1..43.

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 24(1), 9-15 (2018).

Mathematica

Select[Range[3, 25000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[LucasL[2*(2*# - JacobiSymbol[#, 5])] - 3, #] &]

Crossrefs

Cf. A005248, A071904, A339129 (a=3, b=1, k=1).

Cf. A339522 (a=5, b=1), A339523 (a=7, b=1).

Sequence in context: A299392 A299193 A300029 * A160768 A160769 A232352

Adjacent sequences: A339518 A339519 A339520 * A339522 A339523 A339524

Keyword

nonn

Author

Ovidiu Bagdasar, Dec 07 2020

Status

approved