cp's OEIS Frontend

A339517 Odd composite integers m such that A000032(2*m-J(m,5)) == J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

%I A339517 #20 Jul 08 2021 23:24:28
%S A339517 323,377,1001,1183,1729,1891,3827,4181,5777,6601,6721,8149,8841,10877,
%T A339517 11663,13201,13981,15251,17119,17711,18407,19043,23407,25877,26011,
%U A339517 27323,30889,34561,34943,35207,39203,40501,41041
%N A339517 Odd composite integers m such that A000032(2*m-J(m,5)) == J(m,5) (mod m), where J(m,5) is the Jacobi symbol.
%C A339517 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 A339517 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 A339517 Here b=-1, a=1, D=5 and k=2, while V(m) recovers A000032(m) (Lucas numbers).
%D A339517 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
%D A339517 D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
%D A339517 D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
%H A339517 Dorin Andrica and Ovidiu Bagdasar, <a href="https://doi.org/10.3390/math9080838">On Generalized Lucas Pseudoprimality of Level k</a>, Mathematics (2021) Vol. 9, 838.
%H A339517 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 A339517 Select[Range[3, 45000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[LucasL[2*# - JacobiSymbol[#, 5]] - JacobiSymbol[#, 5], #] &]
%Y A339517 Cf. A000032, A071904, A339125 (a=1, b=-1, k=1).
%Y A339517 Cf. A339518 (a=3, b=-1), A339519 (a=5, b=-1), A339520 (a=7, b=-1).
%K A339517 nonn
%O A339517 1,1
%A A339517 _Ovidiu Bagdasar_, Dec 07 2020