A337236 Composite integers m such that A001076(m)^2 == 1 (mod m).

9, 63, 99, 119, 161, 207, 209, 231, 279, 323, 341, 377, 391, 549, 589, 671, 759, 779, 799, 897, 901, 1007, 1159, 1281, 1443, 1449, 1551, 1853, 1891, 2001, 2047, 2071, 2379, 2407, 2501, 2737, 2743, 2849, 2871, 2961, 3069, 3289, 3689, 3827, 4059, 4181, 4199, 4209, 4577

Offset: 1

Author

Ovidiu Bagdasar, Aug 20 2020

Keywords

• nonn

Comments

If p is a prime, then A001076(p)^2==1 (mod p).

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

The generalized Lucas sequence 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^2(p)==1 (mod p) whenever p is prime and b=-1,1.

For a=4, b=-1, U(n) recovers A001076(n).

References

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

Crossrefs

Cf. A337231 (a=1, odd terms), A337232 (a=1, even terms), A337233 (a=2), A337234 (a=3, odd terms), A337235 (a=3, even terms).

Programs

Mathematica

Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 4]*Fibonacci[#, 4] - 1, #] &]