login
A338313
Even composite positive integers m such that A052918(m-1)^2 == 1 (mod m).
0
4, 8, 16, 32, 68, 248, 268, 544, 1328, 4216, 4768, 9112, 9376, 12664, 20128, 22112, 24536, 25544, 30488, 43262, 61574, 125792, 148004, 304792, 398248, 493646, 648848, 913456, 1036664, 1975784, 2350792, 3672454, 4248488, 5422688, 6318188, 6768928, 7079656, 8560724
OFFSET
1,1
COMMENTS
If p is a prime, then A052918(p-1)^2 == 1 (mod p).
This sequence contains the even composite integers for which the congruence holds.
The generalized Lucas sequence of integer parameters (a,b) defined by U(m+2) = a*U(m+1)-b*U(m) 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=5, b=-1, U(m) recovers A052918(m-1), for m=1,2,....
REFERENCES
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)
MATHEMATICA
Select[Range[2, 20000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 5]*Fibonacci[#, 5] - 1, #] &]
CROSSREFS
Cf. A337235 (a=3)
Sequence in context: A049934 A328634 A089890 * A049932 A272712 A020161
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Oct 22 2020
EXTENSIONS
More terms from Amiram Eldar, Oct 22 2020
a(31)-a(38) from Daniel Suteu, Oct 22 2020
STATUS
approved