cp's OEIS Frontend

A337627 Odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 4 (mod m), where U(m) and V(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=4 and b=-1, respectively.

%I A337627 #12 Nov 23 2023 13:17:03
%S A337627 9,161,341,897,901,1281,1853,2737,4181,4209,4577,5473,5611,5777,6119,
%T A337627 6721,9701,9729,10877,11041,12209,12349,13201,13481,14981,15251,16771,
%U A337627 19669,20591,20769,20801,23323,27403,27613,28421,29281,29489,32929,33001,34561,38801
%N A337627 Odd composite integers m such that U(m)^2 == 1 (mod m) and  V(m) == 4 (mod m), where U(m) and V(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=4 and b=-1, respectively.
%C A337627 Intersection of A335670 and A337236.
%C A337627 For a,b integers, the following sequences are defined:
%C A337627 generalized Lucas sequences by U(n+2)=a*U(n+1)-b*U(n) and U(0)=0, U(1)=1,
%C A337627 generalized Pell-Lucas sequences by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a.
%C A337627 These satisfy the identities U(p)^2 == 1 and V(p)==a (mod p) for p prime and b=1,-1.
%C A337627 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 A337627 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 A337627 Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 4]*Fibonacci[#, 4] - 1, #] && Divisible[LucasL[#, 4] - 4, #] &]
%Y A337627 Cf. A335670 and A337236. Similar sequences: A337625 (a=1), A337626 (a=3).
%K A337627 nonn
%O A337627 1,1
%A A337627 _Ovidiu Bagdasar_, Sep 19 2020
%E A337627 More terms from _Amiram Eldar_, Sep 19 2020