%I #36 Nov 23 2023 12:02:34

%S 10,209,230,231,399,430,455,530,901,903,923,989,1295,1729,1855,2015,

%T 2211,2345,2639,2701,2795,2911,3007,3201,3439,3535,3801,4823,5291,

%U 5719,6061,6767,6989,7421,8569,9503,9591,9869,9890,10439,10609,11041,11395,11951,11991

%N Composite integers m such that A003500(m) == 4 (mod m).

%C If p is a prime, then A003500(p)==4 (mod p).

%C This sequence contains the composite integers for which the congruence holds.

%C The generalized Pell-Lucas sequences 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)=A003500(n).

%D D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).

%H Amiram Eldar, <a href="/A335673/b335673.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 odd terms from Chai Wah Wu)

%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 m=10 is the first composite integer for which A003500(m)==4 (mod m).

%t Select[Range[3, 20000], CompositeQ[#] && Divisible[Round@LucasL[2#, Sqrt[2]] - 4, #] &] (* _Amiram Eldar_, Jun 18 2020 *)

%o (PARI) my(M=[1,2;1,3]); forcomposite(m=5, 10^5, if(trace(Mod(M,m)^m)==4, print1(m,", "))); \\ _Joerg Arndt_, Jun 18 2020

%Y Cf. A005248, A335669 (a=3,b=-1), A335672 (a=3,b=1), A335674 (a=5,b=1).

%Y A330206 is the subsequence of odd terms.

%K nonn

%O 1,1

%A _Ovidiu Bagdasar_, Jun 17 2020

%E More terms from _Joerg Arndt_, Jun 18 2020