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%I #10 Nov 23 2023 13:18:04
%S 9,63,99,119,161,207,209,231,279,323,341,377,391,549,589,671,759,779,
%T 799,897,901,1007,1159,1281,1443,1449,1551,1853,1891,2001,2047,2071,
%U 2379,2407,2501,2737,2743,2849,2871,2961,3069,3289,3689,3827,4059,4181,4199,4209,4577
%N Composite integers m such that A001076(m)^2 == 1 (mod m).
%C If p is a prime, then A001076(p)^2==1 (mod p).
%C This sequence contains the composite integers for which the congruence holds.
%C 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.
%C For a=4, b=-1, U(n) recovers A001076(n).
%D D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).
%H D. Andrica and O. Bagdasar, <a href="https://repository.derby.ac.uk/item/92yqq/on-some-new-arithmetic-properties-of-the-generalized-lucas-sequences">On some new arithmetic properties of the generalized Lucas sequences</a>, preprint for Mediterr. J. Math. 18, 47 (2021).
%t Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 4]*Fibonacci[#, 4] - 1, #] &]
%Y Cf. A337231 (a=1, odd terms), A337232 (a=1, even terms), A337233 (a=2), A337234 (a=3, odd terms), A337235 (a=3, even terms).
%K nonn
%O 1,1
%A _Ovidiu Bagdasar_, Aug 20 2020