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.

9, 27, 65, 121, 385, 533, 1035, 4081, 5089, 5993, 6721, 7107, 10877, 11285, 13281, 13741, 14705, 16721, 18901, 19601, 19951, 20705, 24769, 25345, 26599, 26937, 28741, 29161, 32639, 37949, 39185, 39985, 45305, 45451, 49105, 50553, 51085, 52801, 57205, 64297, 72385

Offset: 1

Author

Ovidiu Bagdasar, Sep 19 2020

Keywords

• nonn

Comments

Intersection of A335671 and A337237.

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=5 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. A335671 and A337237.

Similar sequences: A337625 (a=1), A337626 (a=3) and A337627 (a=4).

Programs

Mathematica

Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 5]*Fibonacci[#, 5] - 1, #] && Divisible[LucasL[#, 5] - 5, #] &]

Extensions

More terms from Amiram Eldar, Sep 19 2020