A338007 Odd composite integers m such that A001906(m)^2 == 1 (mod m).

9, 21, 63, 99, 231, 323, 329, 369, 377, 423, 451, 861, 903, 1081, 1189, 1443, 1551, 1819, 1833, 1869, 1891, 2033, 2211, 2737, 2849, 2871, 2961, 3059, 3289, 3653, 3689, 3827, 4059, 4089, 4179, 4181, 4879, 5671, 5777, 6447, 6479, 6601, 6721, 6903, 7743

Offset: 1

Author

Ovidiu Bagdasar, Oct 06 2020

Keywords

• nonn

Comments

For a, b integers, the generalized Lucas sequence is defined by the relation U(n+2)=a*U(n+1)-b*U(n) and U(0)=0, U(1)=1.

This sequence satisfies the relation U(p)^2 == 1 for p prime and b=1,-1.

The composite numbers with this property may be called weak generalized Lucas pseudoprimes of parameters a and b.

The current sequence is defined for a=3 and b=1.

References

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

Links

• Dorin Andrica and Ovidiu Bagdasar, On Generalized Lucas Pseudoprimality of Level k, Mathematics (2021) Vol. 9, 838.
• 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. A338008 (a=4, b=1), A338009 (a=5, b=1), A338010 (a=6, b=1), A338011 (a=7, b=1).

Programs

Mathematica

Select[Range[3, 8000, 2], CompositeQ[#] && Divisible[ChebyshevU[#-1, 3/2]*ChebyshevU[#-1, 3/2] - 1, #] &]