A337629 Odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 6 (mod m), where U(m) and V(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=6 and b=-1, respectively.

57, 481, 629, 721, 779, 1121, 1441, 1729, 2419, 2737, 6721, 7471, 8401, 9361, 10561, 11521, 11859, 12257, 15281, 16321, 16583, 18849, 24721, 25441, 25593, 33649, 35219, 36481, 36581, 37949, 39169, 41041, 45961, 46999, 50681, 52417, 53041, 53521, 54757, 55537

Offset: 1

Author

Ovidiu Bagdasar, Sep 19 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=6 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. A337625 (a=1), A337626 (a=3), A337627 (a=4), A337628 (a=5).

Programs

Mathematica

Select[Range[3, 15000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 6]*Fibonacci[#, 6] - 1, #] && Divisible[LucasL[#, 6] - 6, #] &]

Extensions

More terms from Amiram Eldar, Sep 19 2020