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%I #16 Nov 24 2023 12:06:42
%S 209,455,901,923,989,1295,1729,1855,2015,2345,2639,2701,2795,2911,
%T 3007,3439,3535,4823,5291,5719,6061,6767,6989,7421,8569,9503,9869,
%U 10439,10609,11041,11395,11951,13133,13529,13735,13871,14701,14839,15505,15841,17119,17815
%N Odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 4 (mod m), where U(m)=A001353(m) and V(m)=A003500(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=4 and b=1, respectively.
%C For a, b integers, the following sequences are defined:
%C generalized Lucas sequences by U(n+2)=a*U(n+1)-b*U(n) and U(0)=0, U(1)=1,
%C generalized Pell-Lucas sequences by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a.
%C These satisfy the identities U(p)^2 == 1 and V(p)==a (mod p) for p prime and b=1,-1.
%C These numbers may be called weak generalized Lucas-Bruckner pseudoprimes of parameters a and b.The current sequence is defined for a=4 and b=1.
%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).
%t Select[Range[3, 10000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 2] - 4, #] && Divisible[ChebyshevU[#-1, 2]*ChebyshevU[#-1, 2] - 1, #] &]
%Y Similar sequences: A337627 (a=4, b=-1).
%K nonn
%O 1,1
%A _Ovidiu Bagdasar_, Sep 20 2020
%E More terms from _Amiram Eldar_, Sep 21 2020
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