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%I #20 Nov 23 2023 12:03:05
%S 9,85,161,341,705,897,901,1105,1281,1853,2465,2737,3745,4181,4209,
%T 4577,5473,5611,5777,6119,6721,9701,9729,10877,11041,12209,12349,
%U 13201,13481,14981,15251,16185,16545,16771,19669,20591,20769,20801,21845,23323,24465,25345
%N Odd composite integers m such that A014448(m) == 4 (mod m).
%C If p is a prime, then A014448(p)==4 (mod p).
%C This sequence contains the odd composite integers for which the congruence holds.
%C The generalized Pell-Lucas sequence of integer parameters (a,b) defined by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a, satisfy the identity V(p)==a (mod p) whenever p is prime and b=-1,1.
%C For a=4, b=-1, V(n) recovers A014448(n) (even Lucas numbers).
%D D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).
%H Chai Wah Wu, <a href="/A335670/b335670.txt">Table of n, a(n) for n = 1..1000</a>
%H D. Andrica and O. Bagdasar, <a href="https://repository.derby.ac.uk/item/92yqq/on-some-new-arithmetic-properties-of-the-generalized-lucas-sequences">On some new arithmetic properties of the generalized Lucas sequences</a>, preprint for Mediterr. J. Math. 18, 47 (2021).
%e 9 is the first odd composite integer for which A014448(9)=439204==4 (mod 9).
%p M:= <<4|1>,<1|0>>:
%p f:= proc(n) uses LinearAlgebra:-Modular;
%p local A;
%p A:= Mod(n,M,integer[8]);
%p A:= MatrixPower(n,A,n);
%p 2*A[1,1] - 4*A[1,2] mod n;
%p end proc:
%p select(t -> f(t) = 4 and not isprime(t), [seq(i,i=3..10^5,2)]); # _Robert Israel_, Jun 19 2020
%t Select[Range[3, 25000, 2], CompositeQ[#] && Divisible[LucasL[3#] - 4, #] &] (* _Amiram Eldar_, Jun 18 2020 *)
%Y Cf. A006497, A005845 (a=1), A330276 (a=2), A335669 (a=3), A335671 (a=5).
%K nonn
%O 1,1
%A _Ovidiu Bagdasar_, Jun 17 2020
%E More terms from _Jinyuan Wang_, Jun 17 2020
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