cp's OEIS Frontend

A339521 Odd composite integers m such that A005248(2*m-J(m,5)) == 3 (mod m), where J(m,5) is the Jacobi symbol.

%I A339521 #14 Dec 15 2020 10:23:56
%S A339521 21,203,323,329,377,451,609,861,1001,1081,1183,1547,1729,1819,1891,
%T A339521 2033,2211,2821,3081,3549,3653,3827,4089,4181,4669,5671,5777,5887,
%U A339521 6601,6721,8149,8557,8841,10877,11309,11663,13201,13601,13861,13981,14701,15051,15251
%N A339521 Odd composite integers m such that A005248(2*m-J(m,5)) == 3 (mod m), where J(m,5) is the Jacobi symbol.
%C A339521 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) (mod p) whenever p is prime, k is a positive integer, b=1 and D=a^2-4.
%C A339521 The composite integers m with the property V(k*m-J(m,D)) == V(k-1) (mod m) are called generalized Pell-Lucas pseudoprimes of level k+ and parameter a.
%C A339521 Here b=1, a=3, D=5 and k=2, while V(m) recovers A005248(m).
%D A339521 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
%D A339521 D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
%D A339521 D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
%H A339521 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 A339521 Select[Range[3, 25000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[LucasL[2*(2*# - JacobiSymbol[#, 5])] - 3, #] &]
%Y A339521 Cf. A005248, A071904, A339129 (a=3, b=1, k=1).
%Y A339521 Cf. A339522 (a=5, b=1), A339523 (a=7, b=1).
%K A339521 nonn
%O A339521 1,1
%A A339521 _Ovidiu Bagdasar_, Dec 07 2020