A339727 Odd composite integers m such that A086902(3*m-J(m,53)) == 51*J(m,53) (mod m), where J(m,53) is the Jacobi symbol.

9, 25, 49, 51, 69, 91, 105, 143, 145, 153, 185, 221, 225, 325, 339, 391, 425, 441, 481, 637, 645, 705, 805, 833, 897, 925, 1001, 1173, 1189, 1207, 1225, 1281, 1299, 1365, 1541, 1633, 1653, 1785, 1813, 1921, 2325, 2599, 2651, 2769, 3133, 3333, 3381, 3605, 3825, 3897

Offset

1,1

Comments

The generalized Pell-Lucas sequences of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy V(k*p-J(p,D)) == V(k-1)*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 V(k*m-J(m,D)) == V(k-1)*J(m,D) (mod m) are called generalized Pell-Lucas pseudoprimes of level k- and parameter a.

Here b=-1, a=7, D=53 and k=3, while V(m) recovers A086902(m), with V(2)=51.

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..50.

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 24(1), 9-15 (2018).

Mathematica

Select[Range[3, 4000, 2], CoprimeQ[#, 53] && CompositeQ[#] && Divisible[LucasL[3*# - JacobiSymbol[#, 53], 7] - 51*JacobiSymbol[#, 53], #] &]

Crossrefs

Cf. A086902, A071904, A339128 (a=7, b=-1, k=1), A339520 (a=7, b=-1, k=2).

Cf. A339724 (a=1, b=-1), A339725 (a=3, b=-1), A339726 (a=5, b=-1).

Sequence in context: A051132 A247687 A075026 * A339128 A113659 A325701

Adjacent sequences: A339724 A339725 A339726 * A339728 A339729 A339730

Keyword

nonn

Author

Ovidiu Bagdasar, Dec 14 2020

Status

approved