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).

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, 10, 60, 120, 11, 10, 60, 120, 12, 24, 13, 28, 48, 24, 14, 48, 24, 15, 40, 60, 120, 16, 24, 17, 36, 24, 18, 24, 19, 18, 24, 20, 60, 120, 21, 16, 24, 22, 30, 120, 23, 48, 24

Offset

1,2

Comments

Row n contains recursive mappings of A001175(x) starting with x = n.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10141 (rows n = 1..2500, flattened)

Brennan Benfield and Oliver Lippard, Fixed points of K-Fibonacci sequences, arXiv:2404.08194 [math.NT], 2024.

J. D. Fulton and W. L. Morris, On arithmetical functions related to the Fibonacci numbers, Acta Arithmetica, 16 (1969), 105-110.

Wikipedia, Pisano period

Example

Table begins:
   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;
  10, 60, 120;
  11, 10,  60, 120;
  12, 24;
  ...

Mathematica

q[{0, 1, _}] := False; q[_] := True;
f[k_][{a_, b_, c_}] := {Mod[b, k], Mod[a + b, k], c + 1};
s[1] := 1; s[k_] := s[k] = Which[
  PrimeQ[k] && k > 5, If[
    AnyTrue[PrimitiveRootList[k], Mod[#^2, k] == Mod[# + 1, k] &],
    k - 1,
    NestWhile[f[k], {1, 1, 1}, q][[-1]] ],
  PrimePowerQ[k], NestWhile[f[k], {1, 1, 1}, q][[-1]], True,
    LCM @@ Map[s[#] &, Power @@@ FactorInteger[k] ] ];
Table[Most@ FixedPointList[s[#] &, n], {n, 24}]

Crossrefs

Cf. A001175, A001179, A235249, A235702.

Sequence in context: A115449 A303851 A218542 * A194452 A242516 A282281

Adjacent sequences: A379643 A379644 A379645 * A379647 A379648 A379649

Keyword

nonn,tabf

Author

Michael De Vlieger, Dec 30 2024

Status

approved