A338313 Even composite positive integers m such that A052918(m-1)^2 == 1 (mod m).

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)

Links

Table of n, a(n) for n=1..38.

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

Adjacent sequences: A338310 A338311 A338312 * A338314 A338315 A338316

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