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Numerators of a FRACTRAN program that produces the iterates of the Collatz (or 3x+1) function.
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%I #12 Sep 06 2023 07:29:05

%S 1,136,5,4,26,7,1,33,5,7

%N Numerators of a FRACTRAN program that produces the iterates of the Collatz (or 3x+1) function.

%C These rational numbers (denominators are in A365329) represent a FRACTRAN program that, when started at 2^m, will produce 2^T(m) as the next power of 2, where T(m) is the Collatz or 3x+1 function = (3m+1)/2 if m is odd, m/2 if m is even (A014682).

%C Monks (2002) uses this program to prove Theorem 1 in his paper.

%H J. H. Conway, "FRACTRAN: A Simple Universal Programming Language for Arithmetic", in J. C. Lagarias, ed., <a href="http://www.ams.org/bookstore-getitem/item=mbk-78">The Ultimate Challenge: The 3x+1 Problem</a>, American Mathematical Society, 2010, p. 249, and in T. M. Cover and B. Gopinath, eds, <a href="https://doi.org/10.1007/978-1-4612-4808-8_2">Open Problems in Communication and Computation</a>, Springer, New York, NY, 1987, pp. 4-26.

%H J. C. Lagarias, ed., <a href="http://www.ams.org/bookstore-getitem/item=mbk-78">The Ultimate Challenge: The 3x+1 Problem</a>, American Mathematical Society, 2010, p. 110.

%H Kenneth G. Monks, <a href="https://hal.science/hal-00958971v1">3x+1 Minus the +</a>, Discrete Mathematics and Theoretical Computer Science, 2002, Vol. 5, pp. 47-54.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/FRACTRAN">FRACTRAN</a>.

%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>

%Y Cf. A014682, A365329 (denominators).

%K nonn,frac,fini,full

%O 1,2

%A _Paolo Xausa_, Sep 01 2023