@article{ApplegateLagarias2003,
  author    = {{David Applegate and Jeffrey C. Lagarias}},
  title     = {{Lower bounds for the total stopping time  of $3x+1$ iterates}},
  journal   = {{Math. Comp.}},
  year      = {{2003}},
  volume    = {{72}},
  pages     = {{1035--1049}},
  url       = {{https://arxiv.org/abs/math/0103054}},
  abstract  = {{The 3X+1 function T(n) is (3n+1)/2 if n is odd and n/2 if n is even. The total stopping time \sigma_\infty (n) for a positive integer n is the number of iterations of the 3x+1 function to reach 1 starting from n, and is \infty if 1 is never reached. The 3x+1 conjecture states that this function is finite. We show that infinitely many n have a finite total stopping time with \sigma_\infty(n) > 6.14316 log n. The proof uses a very large computation. It is believed that almost all positive integers have \sigma_\infty (n) > 6.95212 \log n. The method of the paper should extend to prove infinitely many integers have this property, but it would require a much larger computation.}},
  note      = {{MR 2004a:11016}},
}