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A337233
Composite integers m such that P(m)^2 == 1 (mod m), where P(m) is the m-th Pell number A000129(m). Also, odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 6 (mod m), where U(m)=A001109(m) and V(m)=A003499(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=6 and b=1, respectively.
8
35, 119, 169, 385, 741, 779, 899, 935, 961, 1105, 1121, 1189, 1443, 1479, 2001, 2419, 2555, 2915, 3059, 3107, 3383, 3605, 3689, 3741, 3781, 3827, 4199, 4795, 4879, 4901, 5719, 6061, 6083, 6215, 6265, 6441, 6479, 6601, 6895, 6929, 6931, 6965, 7055, 7107, 7801, 8119
OFFSET
1,1
COMMENTS
For a, b integers, the following sequences are defined:
generalized Lucas sequences by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1,
generalized Pell-Lucas sequences by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a.
In general, one has U^2(p) == 1 and V(p)==a (mod p) whenever p is prime and b=1, -1.
The composite numbers satisfying these congruences may be called weak generalized Lucas-Bruckner pseudoprimes of parameters a and b.
For a=2 and b=-1, U(m) recovers A000129(m) (Pell numbers).
For a=6 and b=1, we have U(m)=A001109(m) and V(m)=A003499(m).
This sequence contains the odd composite integers for which the congruence A000129(m)^2 == 1 (mod m) holds.
This is also the sequence of odd composite numbers satisfying the congruences A001109(m)^2 == 1 and A003499(m)==a (mod m).
REFERENCES
D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
LINKS
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[3, 25000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 2]*Fibonacci[#, 2] - 1, #] &]
Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 3] - 6, #] && Divisible[ChebyshevU[#-1, 3]*ChebyshevU[#-1, 3] - 1, #] &]
CROSSREFS
Cf. A337231 (a=1, odd terms), A337232 (a=1, even terms), A337629 (a=6, b=-1), A337778 (a=4, b=1), A337779 (a=5, b=1).
Sequence in context: A044667 A262455 A132144 * A230214 A284876 A216268
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Aug 20 2020
STATUS
approved