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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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..44.

Dorin Andrica, Vlad Cri┼čan, 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: A063653 A222348 A075740 * A129217 A116060 A145054

Adjacent sequences:  A340237 A340238 A340239 * A340241 A340242 A340243

KEYWORD

nonn

AUTHOR

Ovidiu Bagdasar, Jan 01 2021

STATUS

approved

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Last modified August 19 16:39 EDT 2022. Contains 356229 sequences. (Running on oeis4.)