A340121 Odd composite integers m such that A054413(2*m-J(m,53)) == 1 (mod m), where J(m,53) is the Jacobi symbol.

25, 35, 39, 49, 51, 65, 91, 147, 175, 245, 301, 325, 343, 391, 455, 507, 575, 605, 637, 663, 741, 833, 897, 903, 935, 1127, 1205, 1225, 1247, 1295, 1505, 1595, 1633, 1715, 1763, 1775, 1911, 1921, 2107, 2275, 2401, 2407, 2499, 2599, 2651, 3025, 3143, 3185, 3311

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=7, D=53 and k=2, while U(m) is A054413(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..49.

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, 10000, 2], CoprimeQ[#, 53] && CompositeQ[#] && Divisible[Fibonacci[2*#-JacobiSymbol[#, 53], 7] - 1, #] &]

Crossrefs

Cf. A054413, A071904, A340096 (a=7, b=-1, k=1).

Cf. A340118 (a=1, b=-1, k=2), A340119 (a=3, b=-1, k=2), A340120 (a=5, b=-1, k=2).

Sequence in context: A098368 A281370 A101066 * A061442 A049514 A049518

Adjacent sequences: A340118 A340119 A340120 * A340122 A340123 A340124

Keyword

nonn

Author

Ovidiu Bagdasar, Dec 28 2020

Status

approved