A340120 Odd composite integers m such that A052918(2*m-J(m,29)) == 1 (mod m), where J(m,29) is the Jacobi symbol.

9, 15, 25, 27, 45, 75, 91, 121, 125, 135, 143, 147, 175, 225, 275, 325, 375, 441, 483, 625, 675, 735, 755, 1125, 1323, 1547, 1573, 1875, 1935, 2015, 2205, 2275, 2485, 2943, 3025, 3125, 3375, 3575, 3675, 3775, 3843, 4375, 5525, 5625, 5819, 6543, 6615, 6721

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)) == 1 (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=2, 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[2*#-JacobiSymbol[#, 29], 5] - 1, #] &]

Crossrefs

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

Cf. A340118 (a=1, b=-1, k=2), A340119 (a=3, b=-1, k=2), A340121 (a=7, b=-1, k=2).

Sequence in context: A330947 A337237 A036315 * A020154 A079290 A176404

Adjacent sequences: A340117 A340118 A340119 * A340121 A340122 A340123

Keyword

nonn

Author

Ovidiu Bagdasar, Dec 28 2020

Status

approved