A337777 Even composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 3 (mod m), where U(m)=A001906(m) and V(m)=A005248(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=3 and b=1, respectively.

4, 44, 836, 1364, 2204, 7676, 7964, 9164, 11476, 12524, 23804, 31124, 32642, 39556, 73124, 80476, 99644, 110564, 128876, 156484, 192676, 199924, 287804, 295196, 315524, 398924, 542242, 715604, 780044, 934876, 987524, 1050524, 1339516, 1390724, 1891124, 1996796

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=3 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. A337626.

Programs

Mathematica

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

Extensions

More terms from Amiram Eldar, Sep 21 2020