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.

9, 161, 341, 897, 901, 1281, 1853, 2737, 4181, 4209, 4577, 5473, 5611, 5777, 6119, 6721, 9701, 9729, 10877, 11041, 12209, 12349, 13201, 13481, 14981, 15251, 16771, 19669, 20591, 20769, 20801, 23323, 27403, 27613, 28421, 29281, 29489, 32929, 33001, 34561, 38801

Offset: 1

Author

Ovidiu Bagdasar, Sep 19 2020

Keywords

• nonn

Comments

Intersection of A335670 and A337236.

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

Cf. A335670 and A337236. Similar sequences: A337625 (a=1), A337626 (a=3).

Programs

Mathematica

Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 4]*Fibonacci[#, 4] - 1, #] && Divisible[LucasL[#, 4] - 4, #] &]

Extensions

More terms from Amiram Eldar, Sep 19 2020