A337781 -id:A337781 - cp's OEIS frontend

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

Ovidiu Bagdasar, Sep 20 2020

nonn

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.

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

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

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

More terms from Amiram Eldar, Sep 21 2020

A337783 Even composite integers m such that U(m)^2 == 1 (mod m), where U(m)=A004187(m) is the m-th generalized Lucas number of parameters a=7 and b=1.

Original entry on oeis.org

4, 8, 16, 44, 104, 136, 152, 164, 176, 232, 286, 442, 496, 656, 836, 856, 976, 1072, 1364, 1378, 1394, 1804, 1826, 2204, 2248, 2584, 2626, 2684, 2834, 3016, 3268, 3536, 3926, 4264, 4346, 4636, 5084, 5104, 5146, 5662, 7208, 7216, 7384, 7676, 7964, 8294, 8632, 8774, 9164, 9316, 9976

Offset: 1

Ovidiu Bagdasar, Sep 20 2020

nonn

This sequence contains the even composite integers for which the congruence holds.

The generalized Lucas sequences of integer parameters (a,b) defined by U(n+2)=a*U(n+1)-b*U(n) and U(0)=0, U(1)=1, satisfies the identity U^(p)==1 (mod p) whenever p is prime and b=-1,1.

D. Andrica and O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).

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

Cf. A337781 and A337782.

Select[Range[2, 10000, 2], CompositeQ[#] && Divisible[ChebyshevU[#-1, 7/2]*ChebyshevU[#-1, 7/2] - 1, #] &]

More terms from Amiram Eldar, Sep 21 2020

cp's OEIS Frontend

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.

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