Finding patterns in the Collatz 3n + 1 problem

Consider a number n.

If n is even, divide it by two.

If n is odd, multiply it by 3 and add one.

The Collatz conjecture states that by continuing this process: starting with any n, applying this algorithm eventually will converge to 1. This is unproven but so far seems to be true.

The following code finds the number less than a given n which produces the longest chain before converging to one:

function [longest] = collatz (n)
% find the number of steps taken for numbers up to n to converge to one 
% using the 3n + 1 algorithm
% return the number whose chain is longest

distances = zeros(1,n);

for i = 2:n
  
  a = i;
  x = 0;
  dist = 0;
  while( a ~= 1)
    if(a <= columns(distances))
      if(distances(1,a) ~= 0)
        dist += distances(1,a);
        break;
      end
    end
    if(mod(a,2) == 0)
      a = a / 2;
    else
      a = (3 * a) + 1;
    end
    
    dist++;
  end
  distances(1,i) = dist;
end

longest = 1;

for i = 1:n
  if(distances(1,i) > distances(longest))
    longest = i;
  end
end

end

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