A340010 The order of the largest weakly connected component of the Collatz digraph of order n.
1, 2, 2, 3, 3, 3, 3, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 12, 12, 12, 12, 13, 13, 21, 21, 22, 22, 22, 22, 24, 24, 25, 25, 26, 26, 26, 26, 27, 27, 28, 28, 33, 33, 33, 33, 34, 34, 36, 36, 37, 37, 37, 37, 39, 39, 40, 40, 40, 40, 40, 40, 41, 41, 42, 42, 44, 44
Offset: 1
Keywords
Examples
The weakly connected components of the Collatz digraph of order 3 are 1 -> 2 -> 1 and the singleton 3. The order of the largest component is #{1, 2} = 2.
The weakly connected components of the Collatz digraph of order 10 correspond to the following partition of {1, 2, ..., 10}: {1, 2, 3, 4, 5, 6, 8, 10}, {7} and {9}. The order of the largest component is #{1, 2, 3, 4, 5, 6, 8, 10} = 8. Hence, a(10) = 8.
The weakly connected components of the Collatz digraph of order 20 correspond to the partition {1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 16, 20}, {7, 9, 11, 14, 17, 18}, {15} and {19}. The order of the largest component is #{1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 16, 20} = 12. Thus, a(20) = 12.
References
- J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010.
Links
- Sebastian Karlsson, Table of n, a(n) for n = 1..20000
- Lorenzo Sauras Altuzarra, Some arithmetical problems that are obtained by analyzing proofs and infinite graphs, arXiv:2002.03075 [math.NT], 2020.
- Thijs Laarhoven, The 3n+1 conjecture, Eindhoven University of Technology, Bachelor thesis (2009). See also.
- Index entries for sequences related to 3x+1 (or Collatz) problem
Programs
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Python
import networkx as nx def T(n): #A014682 return n//2 if n%2 == 0 else (3*n+1)//2 def a(n): G = nx.Graph() G.add_nodes_from(range(1, n+1)) G.add_edges_from([(m,T(m)) for m in range(1, n+1) if T(m) <= n]) return len(max(nx.connected_components(G))) for n in range(1, 70): print(a(n), end=", ")
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