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A337237 Odd composite integers such that A052918(m-1)^2 == 1 (mod m).

%I A337237 #22 Nov 23 2023 13:17:49
%S A337237 9,15,25,27,35,45,65,75,91,121,135,143,175,225,275,325,385,455,533,
%T A337237 595,615,675,935,1035,1107,1325,1359,1431,1495,1547,1573,1935,2015,
%U A337237 2255,2275,2775,3025,3059,3575,3605,4025,4081,4235,4355,5005,5089,5475,5525,5719,5993,6165
%N A337237 Odd composite integers such that A052918(m-1)^2 == 1 (mod m).
%C A337237 If p is a prime, then A052918(p-1)^2 == 1 (mod p).
%C A337237 This sequence contains the odd composite integers for which the congruence holds.
%C A337237 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 (this property is a form of pseudoprimality).
%C A337237 For a=5, b=-1, U(n) recovers A052918(n-1), for n=1,2,....
%D A337237 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).
%H A337237 Amiram Eldar, <a href="/A337237/b337237.txt">Table of n, a(n) for n = 1..1000</a>
%H A337237 Dorin Andrica and Ovidiu Bagdasar, <a href="https://doi.org/10.3390/math9080838">On Generalized Lucas Pseudoprimality of Level k</a>, Mathematics (2021) Vol. 9, 838.
%H A337237 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 A337237 Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 5]*Fibonacci[#, 5] - 1, #] &]
%Y A337237 Cf. A337231 (a=1, odd terms), A337232 (a=1, even terms), A337233 (a=2), A337234 (a=3, odd terms), A337235 (a=3, even terms), A337236 (a=4).
%K A337237 nonn
%O A337237 1,1
%A A337237 _Ovidiu Bagdasar_, Aug 20 2020