A339519 Odd composite integers m such that A087130(2*m-J(m,29)) == 5*J(m,29) (mod m), where J(m,29) is the Jacobi symbol.

9, 15, 27, 39, 45, 91, 117, 121, 135, 143, 195, 287, 351, 507, 585, 741, 1521, 1547, 1573, 1755, 1935, 2015, 2067, 2535, 2601, 3157, 3227, 3445, 3505, 3519, 3731, 4563, 4879, 4921, 6201, 6273, 6543, 6591, 6721, 7605, 7803, 8099, 10335, 10377, 10403, 10515

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)*J(p,D) (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)*J(m,D) (mod m) are called generalized Pell-Lucas pseudoprimes of level k- and parameter a.

Here b=-1, a=5, D=29 and k=2, while V(m) recovers A087130(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..46.

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, 20000, 2],  CoprimeQ[#, 29] && CompositeQ[#] && Divisible[LucasL[2*# - JacobiSymbol[#, 29], 5] - 5*JacobiSymbol[#, 29], #] &]

Crossrefs

Cf. A087130, A071904, A339127 (a=5, b=-1, k=1).

Cf. A339517 (a=1, b=-1), A339518 (a=3, b=-1), A339520 (a=7, b=-1).

Sequence in context: A013569 A129401 A164385 * A258813 A046353 A340095

Adjacent sequences: A339516 A339517 A339518 * A339520 A339521 A339522

Keyword

nonn

Author

Ovidiu Bagdasar, Dec 07 2020

Status

approved