A340122 Odd composite integers m such that A001906(2*m-J(m,5)) == J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

9, 21, 27, 63, 81, 189, 243, 323, 329, 351, 377, 423, 451, 567, 729, 783, 861, 891, 963, 1081, 1701, 1743, 1819, 1891, 1967, 2033, 2187, 2211, 2871, 2889, 2961, 3321, 3653, 3807, 3827, 4089, 4181, 5103, 5229, 5671, 5777, 5901, 6561, 6601, 6721, 6741, 7587

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=3, D=5 and k=2, 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

Table of n, a(n) for n=1..47.

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[2*#  - JacobiSymbol[#, 5]  - 1, 3/2] - JacobiSymbol[#, 5],  #] &]
Select[Range[3, 10000, 2], CoprimeQ[#, 5] && CompositeQ[#]
&& Divisible[Fibonacci[2*(2*#-JacobiSymbol[#, 5]), 1] - JacobiSymbol[#, 5], #] &]

Crossrefs

Cf. A001906, A071904, A340097 (a=3, b=1, k=1).

Cf. A340123 (a=5, b=1, k=2), A340124 (a=7, b=1, k=2).

Sequence in context: A106214 A348839 A036316 * A339728 A267216 A039289

Adjacent sequences: A340119 A340120 A340121 * A340123 A340124 A340125

Keyword

nonn

Author

Ovidiu Bagdasar, Dec 28 2020

Status

approved