A337782 Even 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.

4, 8, 44, 104, 136, 152, 232, 286, 442, 836, 1364, 1378, 2204, 2584, 2626, 2684, 2834, 3016, 3926, 4636, 5662, 7208, 7384, 7676, 7964, 8294, 9164, 9316, 11476, 12524, 14824, 15224, 17324, 20026, 20474, 21736, 21944, 22814, 23804, 24616, 26596, 27028, 27404, 31124

Offset

1,1

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

Table of n, a(n) for n=1..44.

D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, preprint for Mediterr. J. Math. 18, 47 (2021).

Mathematica

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

Crossrefs

Cf. A337630 (a=7, b=-1), A337777 (a=3, b=1), A337781 (a=7, b=1).

Sequence in context: A163343 A284972 A045639 * A285751 A189538 A351071

Adjacent sequences: A337779 A337780 A337781 * A337783 A337784 A337785

Keyword

nonn

Author

Ovidiu Bagdasar, Sep 20 2020

Extensions

More terms from Amiram Eldar, Sep 21 2020

Status

approved