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A337236
Composite integers m such that A001076(m)^2 == 1 (mod m).
4
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
1,1
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).
MATHEMATICA
Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 4]*Fibonacci[#, 4] - 1, #] &]
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).
Sequence in context: A159235 A181403 A292309 * A285456 A085645 A299579
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Aug 20 2020
STATUS
approved