cp's OEIS Frontend

A337628 Odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 5 (mod m), where U(m) and V(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=5 and b=-1, respectively.

%I A337628 #15 Nov 23 2023 13:16:55
%S A337628 9,27,65,121,385,533,1035,4081,5089,5993,6721,7107,10877,11285,13281,
%T A337628 13741,14705,16721,18901,19601,19951,20705,24769,25345,26599,26937,
%U A337628 28741,29161,32639,37949,39185,39985,45305,45451,49105,50553,51085,52801,57205,64297,72385
%N A337628 Odd composite integers m such that U(m)^2 == 1 (mod m) and  V(m) == 5 (mod m), where U(m) and V(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=5 and b=-1, respectively.
%C A337628 Intersection of A335671 and A337237.
%C A337628 For a,b integers, the following sequences are defined:
%C A337628 generalized Lucas sequences by U(n+2)=a*U(n+1)-b*U(n) and U(0)=0, U(1)=1,
%C A337628 generalized Pell-Lucas sequences by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a.
%C A337628 These satisfy the identities U(p)^2 == 1 and V(p)==a (mod p) for p prime and b=1,-1.
%C A337628 These numbers may be called weak generalized Lucas-Bruckner pseudoprimes of parameters a and b.The current sequence is defined for a=5 and b=-1.
%H A337628 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 A337628 Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 5]*Fibonacci[#, 5] - 1, #] && Divisible[LucasL[#, 5] - 5, #] &]
%Y A337628 Cf. A335671 and A337237.
%Y A337628 Similar sequences: A337625 (a=1), A337626 (a=3) and A337627 (a=4).
%K A337628 nonn
%O A337628 1,1
%A A337628 _Ovidiu Bagdasar_, Sep 19 2020
%E A337628 More terms from _Amiram Eldar_, Sep 19 2020