A337777 -id:A337777 - 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

A338311 Even composites m such that A003499(m)==6 (mod m).

Original entry on oeis.org

4, 14, 28, 164, 434, 574, 1106, 5084, 5572, 7874, 8386, 13454, 13694, 19964, 21988, 33166, 39934, 40132, 95122, 103886, 113918, 148994, 157604, 215326, 216124, 256004, 277564, 306404, 341342, 366148, 571154, 660674, 662494, 764956, 771374, 876644, 981646, 1070926

Offset: 1

Ovidiu Bagdasar, Oct 22 2020

nonn

If p is a prime, then A003499(p)==6 (mod p).
This sequence contains the even composite integers for which the congruence holds.
The generalized Pell-Lucas sequence of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy the identity V(p)==a (mod p) whenever p is prime and b=-1,1.
For a=6 and b=1, V(m) recovers A003499(m).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (2020)
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021)

Cf. A337233 (sequence of odd terms), A337777 (a=3).

Select[Range[2, 25000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 3] - 6, #] &]

More terms from Amiram Eldar, Oct 22 2020

A338312 Even composites m such that A056854(m)==7 (mod m).

Original entry on oeis.org

4, 8, 10, 20, 40, 44, 104, 136, 152, 170, 190, 232, 260, 286, 442, 580, 740, 836, 890, 1364, 1378, 1990, 2204, 2260, 2584, 2626, 2684, 2834, 3016, 3160, 3230, 3926, 4220, 4636, 5662, 6290, 7208, 7384, 7540, 7676, 7964, 8294, 8420, 9164, 9316, 9320, 10070, 11476

Offset: 1

Ovidiu Bagdasar, Oct 22 2020

nonn

If p is a prime, then A056854(p)==7 (mod p).
This sequence contains the even composite integers for which the congruence holds.
The generalized Pell-Lucas sequence of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy the identity V(p)==a (mod p) whenever p is prime and b=-1,1.
For a=7 and b=1, V(m) recovers A056854(m).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (2020)
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021)

Cf. A338082 (sequence of odd terms), A337777 (a=3), A338311 (a=6).

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

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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A338311 Even composites m such that A003499(m)==6 (mod m).

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A338312 Even composites m such that A056854(m)==7 (mod m).

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Mathematica