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A340124
Odd composite integers m such that A004187(2*m-J(m,45)) == J(m,45) (mod m) and gcd(m,45)=1, where J(m,45) is the Jacobi symbol.
3
49, 323, 329, 343, 377, 451, 1081, 1127, 1771, 1819, 1891, 2033, 2303, 2401, 3653, 3827, 4181, 5671, 5777, 6601, 6721, 7471, 7931, 8149, 8557, 9691, 10877, 11309, 11663, 13201, 13861, 13981, 14701, 15251, 15449, 16121, 16807, 17119, 17513, 17687, 17711
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(2*p-J(p,D)) == 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)) == J(m,D)*U(k-1) (mod m) are called generalized Lucas pseudoprimes of level k+ and parameter a.
Here b=1, a=7, D=45 and k=2, while U(m) is A004187(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, 20000, 2], CoprimeQ[#, 45] && CompositeQ[#] &&
Divisible[ ChebyshevU[2*# - JacobiSymbol[#, 45] - 1, 7/2] - JacobiSymbol[#, 45], #] &]
CROSSREFS
Cf. A004187, A071904, A340099 (a=7, b=1, k=1).
Cf. A340122 (a=3, b=1, k=2), A340123 (a=5, b=1, k=2).
Sequence in context: A251222 A250967 A245033 * A017474 A335389 A036318
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Dec 28 2020
STATUS
approved