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A340097 Odd composite integers m such that A001906(m-J(m,5)) == 0 (mod m) and gcd(m,5)=1, where J(m,5) is the Jacobi symbol.

%I A340097 #15 Jul 21 2022 03:09:29
%S A340097 21,323,329,377,451,861,1081,1819,1891,2033,2211,3653,3827,4089,4181,
%T A340097 5671,5777,6601,6721,8149,8557,10877,11309,11663,13201,13861,13981,
%U A340097 14701,15251,17119,17513,17711,17941,18407,19043,19951,20473,23407,25369,25651,25877,27323,27511
%N A340097 Odd composite integers m such that A001906(m-J(m,5)) == 0 (mod m) and gcd(m,5)=1, where J(m,5) is the Jacobi symbol.
%C A340097 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 the identity
%C A340097 U(p-J(p,D)) == 0 (mod p) when p is prime, b=1 and D=a^2-4.
%C A340097 This sequence contains the odd composite integers with U(m-J(m,D)) == 0 (mod m).
%C A340097 For a=3 and b=1, we have D=5 and U(m) recovers A001906(m).
%D A340097 D. Andrica and O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
%H A340097 D. Andrica and O. Bagdasar, <a href="https://doi.org/10.1007/s00009-020-01653-w">On some new arithmetic properties of the generalized Lucas sequences</a>, Mediterr. J. Math., 18, 47 (2021).
%H A340097 D. Andrica and O. Bagdasar, <a href="https://doi.org/10.3390/math9080838">On generalized pseudoprimality of level k</a>, Mathematics 2021, 9(8), 838.
%H A340097 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, 2018, 24(1), 9--15.
%t A340097 Select[Range[3, 30000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[ChebyshevU[# - JacobiSymbol[#, 5] - 1, 3/2], #] &]
%Y A340097 Cf. A001906, A071904, A081264 (a=1, b=-1), A327653 (a=3,b=-1), A340095 (a=5, b=-1), A340096 (a=7, b=-1), A340098 (a=5, b=1), A340099 (a=7, b=1).
%K A340097 nonn
%O A340097 1,1
%A A340097 _Ovidiu Bagdasar_, Dec 28 2020
%E A340097 Coprime condition added to definition by _Georg Fischer_, Jul 20 2022