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A375018 Numbers k such that repeated application of the Pisano period eventually gives 24.

%I A375018 #29 Aug 04 2024 20:50:38
%S A375018 2,3,4,6,7,8,9,12,13,14,16,17,18,19,21,23,24,26,27,28,29,32,34,36,37,
%T A375018 38,39,42,46,47,48,49,51,52,53,54,56,57,58,59,63,64,67,68,69,72,73,74,
%U A375018 76,78,79,81,83,84,87,91,92,94,96,97,98
%N A375018 Numbers k such that repeated application of the Pisano period eventually gives 24.
%C A375018 This sequence is infinite. A number n is a fixed point if the Pisano period of n is equal to n. The trajectory of k is the sequence of values the Pisano period takes on under repeated iteration, starting at k and leading to a fixed point; this sequence is the sequence of integers such that the trajectory leads to 24.
%H A375018 B. Benfield and O. Lippard, <a href="https://arxiv.org/abs/2404.08194">Fixed points of K-Fibonacci Pisano periods</a>, arXiv:2404.08194 [math.NT], 2024.
%H A375018 J. Fulton and W. Morris, <a href="https://eudml.org/doc/204918">On arithmetical functions related to the Fibonacci numbers</a>, Acta Arith., 2(16):105-110, 1969.
%H A375018 E. Trojovska, <a href="https://doi.org/10.3390/math8050773">On periodic points of the order of appearance in the Fibonacci sequence</a>, Mathematics, 2020.
%e A375018 a(1)=2 because 2 is the smallest number with Pisano period trajectory terminating at 24: pi(2)=3, pi(3)=8, pi(8)=12, pi(12)=24.
%o A375018 (Sage)
%o A375018 L=[]
%o A375018 for i in range(2,101):
%o A375018     a=i
%o A375018     y=BinaryRecurrenceSequence(1,1,0,1).period(Integer(i))
%o A375018     while a!=y:
%o A375018         a=y
%o A375018         y=BinaryRecurrenceSequence(1,1,0,1).period(Integer(a))
%o A375018     if a==24:
%o A375018         L.append(i)
%o A375018 print(L)
%Y A375018 Cf. A001175.
%K A375018 nonn
%O A375018 1,1
%A A375018 _Oliver Lippard_ and _Brennan G. Benfield_, Aug 04 2024