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A339521
Odd composite integers m such that A005248(2*m-J(m,5)) == 3 (mod m), where J(m,5) is the Jacobi symbol.
3
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
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
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Dec 07 2020
STATUS
approved