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A340123 Odd composite integers m such that A004254(2*m-J(m,21)) == J(m,21) (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol. 3
25, 115, 125, 253, 275, 391, 425, 505, 527, 551, 575, 625, 713, 715, 775, 779, 935, 1705, 1807, 1919, 2525, 2627, 2875, 2893, 2929, 3125, 3281, 4033, 4141, 5191, 5555, 5671, 5777, 5983, 6049, 6325, 6479, 6565, 6575, 6875, 7625, 7645, 7739, 8585, 8695, 9361, 9451, 9775 (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(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=5, D=21 and k=2, 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..48.

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, 10000, 2], CoprimeQ[#, 21] && CompositeQ[#] &&

Divisible[ ChebyshevU[2*# - JacobiSymbol[#, 21] - 1, 5/2] - JacobiSymbol[#, 21],  #] &]

CROSSREFS

Cf. A004254, A071904, A340098 (a=5, b=1, k=1).

Cf. A340122 (a=3, b=1, k=2), A340124 (a=7, b=1, k=2).

Sequence in context: A044657 A160437 A050589 * A247683 A020152 A218493

Adjacent sequences:  A340120 A340121 A340122 * A340124 A340125 A340126

KEYWORD

nonn

AUTHOR

Ovidiu Bagdasar, Dec 28 2020

STATUS

approved

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Last modified August 10 10:09 EDT 2022. Contains 356039 sequences. (Running on oeis4.)