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URL: https://oeis.org/A340240

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A340240
Odd composite integers m such that A004254(3*m-J(m,21)) == 5*J(m,21) (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.
2
55, 407, 527, 529, 551, 559, 965, 1199, 1265, 1633, 1807, 1919, 1961, 3401, 3959, 4033, 4381, 5461, 5777, 5977, 5983, 6049, 6233, 6439, 6479, 7141, 7195, 7645, 7999, 8639, 8695, 8993, 9265, 9361, 11663, 11989, 12209, 12265, 13019, 13021, 13199, 14023, 14465, 14491
OFFSET
1,1
COMMENTS
The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy U(3*p-J(p,D)) == a*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 U(k*m-J(m,D)) == U(k-1)*J(m,D) (mod m) are called generalized Lucas pseudoprimes of level k+ and parameter a.
Here b=1, a=5, D=21 and k=3, while U(m) is A004254(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, 2018, 24(1), 9--15.
MATHEMATICA
Select[Range[3, 15000, 2], CoprimeQ[#, 21] && CompositeQ[#] && Divisible[ ChebyshevU[3*# - JacobiSymbol[#, 21] - 1, 5/2] - 5*JacobiSymbol[#, 21], #] &]
CROSSREFS
Cf. A004254, A071904, A340098 (a=5, b=1, k=1), A340123 (a=5, b=1, k=2).
Cf. A340239 (a=3, b=1, k=3), A340241 (a=7, b=1, k=3).
Sequence in context: A222348 A075740 A380070 * A355511 A129217 A116060
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Jan 01 2021
STATUS
approved