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A335671 Odd composite integers m such that A087130(m) == 5 (mod m).

%I A335671 #20 Nov 23 2023 12:03:15
%S A335671 9,27,65,121,145,377,385,533,1035,1189,1305,1885,2233,2465,4081,5089,
%T A335671 5993,6409,6721,7107,10877,11281,11285,13281,13369,13741,13833,14705,
%U A335671 15457,16721,17545,18901,19601,19951,20329,20705,22881,24769,25345,26599,26937,28741,29161
%N A335671 Odd composite integers m such that A087130(m) == 5 (mod m).
%C A335671 If p is a prime, then A087130(p)==5 (mod p).
%C A335671 This sequence contains the odd composite integers for which the congruence holds.
%C A335671 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 A335671 For a=5, b=-1, V(n) recovers A087130(n).
%D A335671 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).
%H A335671 Robert Israel, <a href="/A335671/b335671.txt">Table of n, a(n) for n = 1..1000</a>
%H A335671 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 A335671 9 is the first odd composite integer for which A087130(9)=2744420==5 (mod 9).
%p A335671 M:= <<5|1>,<1|0>>:
%p A335671 f:= proc(n) uses LinearAlgebra:-Modular;
%p A335671 local A;
%p A335671 A:= Mod(n,M,integer[8]);
%p A335671 A:= MatrixPower(n,A,n);
%p A335671 2*A[1,1] - 5*A[1,2] mod n;
%p A335671 end proc:
%p A335671 select(t -> f(t) = 5 and not isprime(t), [seq(i,i=3..10^5,2)]); # _Robert Israel_, Jun 19 2020
%t A335671 Select[Range[3, 30000, 2], CompositeQ[#] && Divisible[LucasL[#, 5] - 5, #] &] (* _Amiram Eldar_, Jun 18 2020 *)
%Y A335671 Cf. A006497, A005845 (a=1), A330276 (a=2), A335669 (a=3), A335670 (a=4).
%K A335671 nonn
%O A335671 1,1
%A A335671 _Ovidiu Bagdasar_, Jun 17 2020
%E A335671 More terms from _Jinyuan Wang_, Jun 17 2020