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

Author

Ovidiu Bagdasar, Oct 22 2020

Keywords

• nonn

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)

Crossrefs

Cf. A337235 (a=3)

Programs

Mathematica

Select[Range[2, 20000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 5]*Fibonacci[#, 5] - 1, #] &]

Extensions

More terms from Amiram Eldar, Oct 22 2020

a(31)-a(38) from Daniel Suteu, Oct 22 2020