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

Author

Ovidiu Bagdasar, Oct 22 2020

Keywords

• nonn

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).

Links

• D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (2021).

Crossrefs

Cf. A004254, A337782 (a=3), A337783 (a=7).

Programs

Mathematica

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

Extensions

More terms from Amiram Eldar, Oct 22 2020