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A379646 Irregular triangle T(n,k) where row n contains the trajectory of recursive mappings of A001175(x) starting with x = n and ending at fixed point A235249(n).

%I A379646 #8 Jan 03 2025 02:20:59
%S A379646 1,2,3,8,12,24,3,8,12,24,4,6,24,5,20,60,120,6,24,7,16,24,8,12,24,9,24,
%T A379646 10,60,120,11,10,60,120,12,24,13,28,48,24,14,48,24,15,40,60,120,16,24,
%U A379646 17,36,24,18,24,19,18,24,20,60,120,21,16,24,22,30,120,23,48,24
%N A379646 Irregular triangle T(n,k) where row n contains the trajectory of recursive mappings of A001175(x) starting with x = n and ending at fixed point A235249(n).
%C A379646 Row n contains recursive mappings of A001175(x) starting with x = n.
%H A379646 Michael De Vlieger, <a href="/A379646/b379646.txt">Table of n, a(n) for n = 1..10141</a> (rows n = 1..2500, flattened)
%H A379646 Brennan Benfield and Oliver Lippard, <a href="https://arxiv.org/abs/2404.08194">Fixed points of K-Fibonacci sequences</a>, arXiv:2404.08194 [math.NT], 2024.
%H A379646 J. D. Fulton and W. L. Morris, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa16/aa1621.pdf">On arithmetical functions related to the Fibonacci numbers</a>, Acta Arithmetica, 16 (1969), 105-110.
%H A379646 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pisano_period">Pisano period</a>
%e A379646 Table begins:
%e A379646    1;
%e A379646    2,  3,   8,  12, 24;
%e A379646    3,  8,  12,  24;
%e A379646    4,  6,  24;
%e A379646    5, 20,  60, 120;
%e A379646    6, 24;
%e A379646    7, 16,  24;
%e A379646    8, 12,  24;
%e A379646    9, 24;
%e A379646   10, 60, 120;
%e A379646   11, 10,  60, 120;
%e A379646   12, 24;
%e A379646   ...
%t A379646 q[{0, 1, _}] := False; q[_] := True;
%t A379646 f[k_][{a_, b_, c_}] := {Mod[b, k], Mod[a + b, k], c + 1};
%t A379646 s[1] := 1; s[k_] := s[k] = Which[
%t A379646   PrimeQ[k] && k > 5, If[
%t A379646     AnyTrue[PrimitiveRootList[k], Mod[#^2, k] == Mod[# + 1, k] &],
%t A379646     k - 1,
%t A379646     NestWhile[f[k], {1, 1, 1}, q][[-1]] ],
%t A379646   PrimePowerQ[k], NestWhile[f[k], {1, 1, 1}, q][[-1]], True,
%t A379646     LCM @@ Map[s[#] &, Power @@@ FactorInteger[k] ] ];
%t A379646 Table[Most@ FixedPointList[s[#] &, n], {n, 24}]
%Y A379646 Cf. A001175, A001179, A235249, A235702.
%K A379646 nonn,tabf
%O A379646 1,2
%A A379646 _Michael De Vlieger_, Dec 30 2024