cp's OEIS Frontend

A337777 Even composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 3 (mod m), where U(m)=A001906(m) and V(m)=A005248(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=3 and b=1, respectively.

%I A337777 #24 Nov 23 2023 13:16:22
%S A337777 4,44,836,1364,2204,7676,7964,9164,11476,12524,23804,31124,32642,
%T A337777 39556,73124,80476,99644,110564,128876,156484,192676,199924,287804,
%U A337777 295196,315524,398924,542242,715604,780044,934876,987524,1050524,1339516,1390724,1891124,1996796
%N A337777 Even composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 3 (mod m), where U(m)=A001906(m) and V(m)=A005248(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=3 and b=1, respectively.
%C A337777 For a, b integers, the following sequences are defined:
%C A337777 generalized Lucas sequences by U(n+2)=a*U(n+1)-b*U(n) and U(0)=0, U(1)=1;
%C A337777 generalized Pell-Lucas sequences by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a.
%C A337777 These satisfy the identities U(p)^2 == 1 and V(p)==a (mod p) for p prime and b=1,-1.
%C A337777 These numbers may be called weak generalized Lucas-Bruckner pseudoprimes of parameters a and b. The current sequence is defined for a=3 and b=1.
%H A337777 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 A337777 Select[Range[2, 20000, 2], CompositeQ[#] && Divisible[LucasL[2#] - 3, #] && Divisible[ChebyshevU[#-1, 3/2]*ChebyshevU[#-1, 3/2] - 1, #] &]
%Y A337777 Cf. A337626.
%K A337777 nonn
%O A337777 1,1
%A A337777 _Ovidiu Bagdasar_, Sep 20 2020
%E A337777 More terms from _Amiram Eldar_, Sep 21 2020