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A337630 Odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 7 (mod m), where U(m) and V(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=7 and b=-1, respectively.

%I A337630 #15 Nov 23 2023 13:16:33
%S A337630 25,51,91,161,325,425,561,791,1105,1633,1921,2001,2465,2599,2651,2737,
%T A337630 7345,8449,9361,10325,10465,10825,11285,12025,12291,13021,15457,17111,
%U A337630 18193,18881,19307,20705,20833,21931,24081,24661,31521,32305,37925,38801,39059,40641
%N A337630 Odd composite integers m such that U(m)^2 == 1 (mod m) and  V(m) == 7 (mod m), where U(m) and V(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=7 and b=-1, respectively.
%C A337630 For a,b integers, the following sequences are defined:
%C A337630 generalized Lucas sequences by U(n+2)=a*U(n+1)-b*U(n) and U(0)=0, U(1)=1,
%C A337630 generalized Pell-Lucas sequences by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a.
%C A337630 These satisfy the identities U(p)^2 == 1 and V(p)==a (mod p) for p prime and b=1,-1.
%C A337630 These numbers may be called weak generalized Lucas-Bruckner pseudoprimes of parameters a and b. The current sequence is defined for a=7 and b=-1.
%H A337630 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 A337630 Select[Range[3, 15000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 7]*Fibonacci[#, 7] - 1, #] && Divisible[LucasL[#, 7] - 7, #] &]
%Y A337630 Cf. A337625 (a=1), A337626 (a=3), A337627 (a=4), A337628 (a=5), A337629 (a=6).
%K A337630 nonn
%O A337630 1,1
%A A337630 _Ovidiu Bagdasar_, Sep 19 2020
%E A337630 More terms from _Amiram Eldar_, Sep 19 2020