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A337629 Odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 6 (mod m), where U(m) and V(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=6 and b=-1, respectively.

%I A337629 #15 Nov 23 2023 13:16:43
%S A337629 57,481,629,721,779,1121,1441,1729,2419,2737,6721,7471,8401,9361,
%T A337629 10561,11521,11859,12257,15281,16321,16583,18849,24721,25441,25593,
%U A337629 33649,35219,36481,36581,37949,39169,41041,45961,46999,50681,52417,53041,53521,54757,55537
%N A337629 Odd composite integers m such that U(m)^2 == 1 (mod m) and  V(m) == 6 (mod m), where U(m) and V(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=6 and b=-1, respectively.
%C A337629 For a,b integers, the following sequences are defined:
%C A337629 generalized Lucas sequences by U(n+2)=a*U(n+1)-b*U(n) and U(0)=0, U(1)=1,
%C A337629 generalized Pell-Lucas sequences by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a.
%C A337629 These satisfy the identities U(p)^2 == 1 and V(p)==a (mod p) for p prime and b=1,-1.
%C A337629 These numbers may be called weak generalized Lucas-Bruckner pseudoprimes of parameters a and b. The current sequence is defined for a=6 and b=-1.
%H A337629 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 A337629 Select[Range[3, 15000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 6]*Fibonacci[#, 6] - 1, #] && Divisible[LucasL[#, 6] - 6, #] &]
%Y A337629 Cf. A337625 (a=1), A337626 (a=3), A337627 (a=4), A337628 (a=5).
%K A337629 nonn
%O A337629 1,1
%A A337629 _Ovidiu Bagdasar_, Sep 19 2020
%E A337629 More terms from _Amiram Eldar_, Sep 19 2020