A340237 - OEIS

A340237

Odd composite integers m such that A052918(3*m-J(m,29)) == 5 (mod m), where J(m,29) is the Jacobi symbol.

9, 27, 33, 35, 65, 81, 99, 121, 221, 243, 297, 363, 513, 585, 627, 705, 729, 891, 1089, 1539, 1541, 1881, 2145, 2187, 2299, 2673, 3267, 3605, 4181, 4573, 4579, 5265, 5633, 6721, 6993, 7865, 8019, 8979, 9131, 9801, 10307, 10877, 10881, 13333, 13741, 14001, 14705, 14989

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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 (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) (mod m) are called generalized Lucas pseudoprimes of level k- and parameter a.

Here b=-1, a=5, D=29 and k=3, while U(m) is A052918(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 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, 15000, 2], CoprimeQ[#, 29] && CompositeQ[#] && Divisible[Fibonacci[3*#-JacobiSymbol[#, 29], 5] - 5, #] &]

CROSSREFS

Cf. A052918, A071904, A340095 (a=5, b=-1, k=1), A340120 (a=5, b=-1, k=2).

Cf. A340235 (a=1, b=-1, k=3), A340236 (a=3, b=-1, k=3), A340238 (a=7, b=-1, k=3).

Sequence in context: A276003 A255343 A108107 * A216168 A036303 A359030

Adjacent sequences: A340234 A340235 A340236 * A340238 A340239 A340240

KEYWORD

nonn

AUTHOR

Ovidiu Bagdasar, Jan 01 2021

STATUS

approved