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A340239
Odd composite integers m such that A001906(3*m-J(m,5)) == 3*J(m,5) (mod m), where J(m,5) is the Jacobi symbol.
2
9, 49, 63, 141, 161, 207, 323, 341, 377, 441, 671, 901, 1007, 1127, 1281, 1449, 1853, 1891, 2071, 2303, 2407, 2501, 2743, 2961, 3827, 4181, 4623, 5473, 5611, 5777, 6119, 6593, 6601, 6721, 7161, 7567, 8149, 8473, 8961, 9729, 9881
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=3, D=5 and k=3, while U(m) is A001906(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, 10000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[ ChebyshevU[3*# - JacobiSymbol[#, 5] - 1, 3/2] - 3*JacobiSymbol[#, 5], #] &]
CROSSREFS
Cf. A001906, A071904, A340097 (a=3, b=1, k=1), A340122 (a=3, b=1, k=2).
Cf. A340240 (a=5, b=1, k=3), A340241 (a=7, b=1, k=3).
Sequence in context: A262537 A133478 A339129 * A032589 A137175 A028375
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Jan 01 2021
STATUS
approved