A337778 Odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 4 (mod m), where U(m)=A001353(m) and V(m)=A003500(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=4 and b=1, respectively.

209, 455, 901, 923, 989, 1295, 1729, 1855, 2015, 2345, 2639, 2701, 2795, 2911, 3007, 3439, 3535, 4823, 5291, 5719, 6061, 6767, 6989, 7421, 8569, 9503, 9869, 10439, 10609, 11041, 11395, 11951, 13133, 13529, 13735, 13871, 14701, 14839, 15505, 15841, 17119, 17815

Offset: 1

Author

Ovidiu Bagdasar, Sep 20 2020

Keywords

• nonn

Comments

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

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

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

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

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.

Links

• D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, preprint for Mediterr. J. Math. 18, 47 (2021).

Crossrefs

Similar sequences: A337627 (a=4, b=-1).

Programs

Mathematica

Select[Range[3, 10000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 2] - 4, #] && Divisible[ChebyshevU[#-1, 2]*ChebyshevU[#-1, 2] - 1, #] &]

Extensions

More terms from Amiram Eldar, Sep 21 2020