A337235 Even composite integers m such that A006190(m)^2 == 1 (mod m).

4, 8, 16, 68, 1208, 1424, 3056, 3824, 3928, 20912, 52174, 63716, 88708, 123148, 161872, 582224, 887566, 17083292, 18900412, 34648888, 39991684, 44884912, 51390736, 103170448, 107825236, 132238514, 279900272, 686071244, 769252508, 3251623346, 3358311986, 3535011826

Offset

1,1

Comments

If p is a prime, then A006190(p)^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(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=3, b=-1, U(n) recovers A006190(n) ("Bronze" Fibonacci numbers).

References

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).

Links

Table of n, a(n) for n=1..32.

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, 25000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 5]*Fibonacci[#, 5] - 1, #] &]

Crossrefs

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

Sequence in context: A230112 A023376 A242966 * A038110 A241197 A130436

Adjacent sequences: A337232 A337233 A337234 * A337236 A337237 A337238

Keyword

nonn

Author

Ovidiu Bagdasar, Aug 20 2020

Extensions

More terms from Amiram Eldar, Aug 21 2020

a(18)-a(32) from Daniel Suteu, Aug 29 2020

Status

approved