cp's OEIS Frontend

A339725 Odd composite integers m such that A006497(3*m-J(m,13)) == 11*J(m,13) (mod m), where J(m,13) is the Jacobi symbol.

%I A339725 #8 Dec 16 2020 05:53:00
%S A339725 9,27,119,133,145,165,205,261,341,393,649,693,705,901,945,1121,1173,
%T A339725 1189,1353,1431,1485,1881,2133,2805,3201,3605,3745,4187,5173,5461,
%U A339725 5841,5945,6165,6213,6485,6943,6993,7107,7991,8321,8449,9669,11041,11781,11961,12861
%N A339725 Odd composite integers m such that A006497(3*m-J(m,13)) == 11*J(m,13) (mod m), where J(m,13) is the Jacobi symbol.
%C A339725 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.
%C A339725 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.
%C A339725 Here b=-1, a=3, D=13 and k=3, while V(m) recovers A006497(m), with V(2)=11.
%D A339725 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
%D A339725 D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
%D A339725 D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
%H A339725 Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, <a href="https://doi.org/10.1016/j.ajmsc.2017.06.002">On Fibonacci and Lucas sequences modulo a prime and primality testing</a>, Arab Journal of Mathematical Sciences, 24(1), 9-15 (2018).
%t A339725 Select[Range[3, 13000, 2], CoprimeQ[#, 13] && CompositeQ[#] && Divisible[LucasL[3*# - JacobiSymbol[#, 13], 3] - 11*JacobiSymbol[#, 13], #] &]
%Y A339725 Cf. A006497, A071904, A339126 (a=3, b=-1, k=1), A339518 (a=3, b=-1, k=2).
%Y A339725 Cf. A339724 (a=1, b=-1), A339726 (a=5, b=-1), A339727 (a=7, b=-1).
%K A339725 nonn
%O A339725 1,1
%A A339725 _Ovidiu Bagdasar_, Dec 14 2020