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

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, 38, 39, 42, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 63, 64, 67, 68, 69, 72, 73, 74, 76, 78, 79, 81, 83, 84, 87, 91, 92, 94, 96, 97, 98

Offset: 1

Author

Oliver Lippard and Brennan G. Benfield, Aug 04 2024

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• B. Benfield and O. Lippard, Fixed points of K-Fibonacci Pisano periods, arXiv:2404.08194 [math.NT], 2024.
• J. Fulton and W. Morris, On arithmetical functions related to the Fibonacci numbers, Acta Arith., 2(16):105-110, 1969.
• E. Trojovska, On periodic points of the order of appearance in the Fibonacci sequence, Mathematics, 2020.

Crossrefs

Cf. A001175.

Programs

Sage

L=[]
for i in range(2,101):
    a=i
    y=BinaryRecurrenceSequence(1,1,0,1).period(Integer(i))
    while a!=y:
        a=y
        y=BinaryRecurrenceSequence(1,1,0,1).period(Integer(a))
    if a==24:
        L.append(i)
print(L)