A338314 Even composite integers m such that A004254(m)^2 == 1 (mod m).

4, 8, 76, 104, 116, 296, 872, 1112, 1378, 2204, 2774, 2834, 3016, 4472, 5174, 5624, 6364, 6554, 8854, 9164, 9976, 10564, 11026, 11324, 11476, 12644, 14356, 14456, 15124, 15544, 15688, 16484, 20492, 20786, 21944, 26506, 26564, 30302, 31996, 32264, 33368, 35048

Offset

1,1

Comments

If p is a prime, then A004254(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(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(n) recovers A004254(m).

These numbers may be called weak generalized Lucas pseudoprimes of parameters a and b. The current sequence is defined for a=5 and b=1.

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..42.

Mathematica

Select[Range[2, 15000, 2], CompositeQ[#] && Divisible[ChebyshevU[#-1, 5/2]*ChebyshevU[#-1, 5/2] - 1, #] &]

Crossrefs

Cf. A337782 (a=3), A337783 (a=7).

Sequence in context: A206346 A240503 A221484 * A288956 A353032 A236284

Adjacent sequences: A338311 A338312 A338313 * A338315 A338316 A338317

Keyword

nonn

Author

Ovidiu Bagdasar, Oct 22 2020

Extensions

More terms from Amiram Eldar, Oct 22 2020

Status

approved