A337781 Odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 7 (mod m), where U(m)=A004187(m) and V(m)=A056854(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=7 and b=1, respectively.

323, 329, 377, 451, 1081, 1189, 1819, 1891, 2033, 2737, 2849, 3059, 3289, 3653, 3689, 3827, 4181, 4879, 5671, 5777, 6479, 6601, 6721, 8149, 8533, 8557, 8569, 8651, 8701, 10199, 10877, 11309, 11339, 11521, 11663, 12341, 13201, 13489, 13861, 13981, 14701, 15251, 15301

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=7 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. A337630 (a=7, b=-1), A337778 (a=4, b=1), A337779 (a=5, b=1), A337780 (a=6, b=1).

Programs

Mathematica

Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 7/2] - 7, #] && Divisible[ChebyshevU[#-1, 7/2]*ChebyshevU[#-1, 7/2] - 1, #] &]