A339730 Odd composite integers m such that A056854(3*m-J(m,45)) == 47 (mod m) and gcd(m,45)=1, where J(m,45) is the Jacobi symbol.

49, 161, 287, 323, 329, 341, 377, 451, 671, 737, 901, 1007, 1079, 1081, 1127, 1271, 1363, 1541, 1819, 1853, 1891, 1927, 2033, 2071, 2303, 2407, 2431, 2461, 2501, 2567, 2743, 3653, 3827, 4181, 4843, 5029, 5243, 5473, 5611, 5671, 5777, 6119, 6593, 6601, 6721, 6923

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=7, D=45 and k=3, while V(m) recovers A056854(m), with V(2)=47.

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

Amiram Eldar, Table of n, a(n) for n = 1..1000

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

Mathematica

Select[Range[3, 7000, 2], CoprimeQ[#, 45] && CompositeQ[#] && Divisible[LucasL[4*(3*# - JacobiSymbol[#, 45])] - 47, #] &]

Crossrefs

Cf. A056854, A071904, A339131 (a=7, b=1, k=1), A339523 (a=7, b=1, k=2).

Cf. A339728 (a=3, b=1), A339729 (a=5, b=1).

Sequence in context: A159247 A265423 A226146 * A338011 A066551 A134210

Adjacent sequences: A339727 A339728 A339729 * A339731 A339732 A339733

Keyword

nonn

Author

Ovidiu Bagdasar, Dec 14 2020

Status

approved