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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 (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=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

Table of n, a(n) for n=1..41.

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[#, 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

Adjacent sequences:  A340236 A340237 A340238 * A340240 A340241 A340242

KEYWORD

nonn

AUTHOR

Ovidiu Bagdasar, Jan 01 2021

STATUS

approved

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Last modified July 2 16:04 EDT 2022. Contains 355029 sequences. (Running on oeis4.)