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A337625 Odd composite integers m such that F(m)^2 == 1 (mod m) and L(m) == 1 (mod m), where F(m) and L(m) are the m-th Fibonacci and Lucas numbers, respectively.

%I A337625 #21 Mar 20 2021 16:03:48
%S A337625 2737,4181,5777,6721,10877,13201,15251,29281,34561,51841,64079,64681,
%T A337625 67861,68251,75077,80189,90061,96049,97921,100127,105281,113573,
%U A337625 118441,146611,161027,162133,163081,179697,186961,194833,197209,219781,228241,231703,252601,254321
%N A337625 Odd composite integers m such that F(m)^2 == 1 (mod m) and  L(m) == 1 (mod m), where F(m) and L(m) are the m-th Fibonacci and Lucas numbers, respectively.
%C A337625 Intersection of A005845 and A337231.
%C A337625 These numbers may be called weak generalized Fibonacci-Lucas-Bruckner pseudoprimes.
%C A337625 If p is a prime, then F(p)^2 == 1 (mod p) and L(p) == 1 (mod p).
%C A337625 This sequence contains the odd composite integers for which these congruences hold.
%C A337625 For a,b integers, the following sequences are defined:
%C A337625 generalized Lucas sequences by U(n+2)=a*U(n+1)-b*U(n) and U(0)=0, U(1)=1,
%C A337625 generalized Pell-Lucas sequences by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a.
%C A337625 These satisfy the identities U(p)^2 == 1 and V(p)==a (mod p) for p prime and b=1,-1.
%C A337625 These numbers may be called weak generalized Lucas-Bruckner pseudoprimes of parameters a and b.The current sequence is defined for a=1 and b=-1.
%C A337625 Examples: a(n) is also the number of Jones graphs on n nodes.
%H A337625 Dorin Andrica and Ovidiu Bagdasar, <a href="https://doi.org/10.1007/s00009-020-01653-w">On some new arithmetic properties of the generalized Lucas sequences</a>, Mediterr. J. Math., 18 (2021), 47.
%t A337625 Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 1]*Fibonacci[#, 1] - 1, #] && Divisible[LucasL[#, 1] - 1, #] &]
%Y A337625 Cf. A005845 and A337231.
%K A337625 nonn
%O A337625 1,1
%A A337625 _Ovidiu Bagdasar_, Sep 19 2020
%E A337625 More terms from _Amiram Eldar_, Sep 19 2020