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URL: https://oeis.org/A339520

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A339520
Odd composite integers m such that A086902(2*m-J(m,53)) == 7*J(m,53) (mod m), where J(m,53) is the Jacobi symbol.
4
25, 35, 51, 65, 75, 91, 105, 175, 203, 325, 391, 455, 575, 645, 861, 1247, 1275, 1295, 1633, 1763, 1775, 1785, 1875, 1921, 2275, 2407, 2415, 2599, 2625, 2651, 3045, 3367, 4199, 4579, 4623, 5629, 5835, 5887, 6441, 6699, 9959, 10465, 10815, 10825, 10877, 11865, 12025
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=7, D=53 and k=2, while V(m) recovers A086902(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
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[#, 53] && CompositeQ[#] && Divisible[LucasL[2*# - JacobiSymbol[#, 53], 7] - 7*JacobiSymbol[#, 53], #] &]
CROSSREFS
Cf. A086902, A071904, A339128 (a=7, b=-1, k=1).
Cf. A339517 (a=1, b=-1), A339518 (a=3, b=-1), A339529 (a=5, b=-1).
Sequence in context: A060976 A391334 A036320 * A340096 A348423 A276290
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Dec 07 2020
STATUS
approved