A337779 - OEIS

%I #17 Nov 24 2023 12:06:52

%S 527,551,1105,1807,1919,2015,2071,2915,3289,4031,4033,4355,5291,5777,

%T 5983,6049,6061,6479,6785,7645,8695,9361,9889,11285,11663,11951,12209,

%U 12265,12545,13079,14491,16211,17119,17249,18299,18407,20087,20099,20845,21505,22499

%N Odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 5 (mod m), where U(m)=A004254(m) and V(m)=A003501(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=5 and b=1, respectively.

%C For a, b integers, the following sequences are defined:

%C generalized Lucas sequences by U(n+2)=a*U(n+1)-b*U(n) and U(0)=0, U(1)=1,

%C generalized Pell-Lucas sequences by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a.

%C These satisfy the identities U(p)^2 == 1 and V(p)==a (mod p) for p prime and b=1,-1.

%C 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 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 Select[Range[3, 10000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 5/2] - 5, #] && Divisible[ChebyshevU[#-1, 5/2]*ChebyshevU[#-1, 5/2] - 1, #] &]

%Y Cf. A337628 (a=5, b=-1), A337778 (a=4, b=1).

%K nonn

%O 1,1

%A _Ovidiu Bagdasar_, Sep 20 2020

%E More terms from _Amiram Eldar_, Sep 21 2020