FRACTRAN (esolang, appeared 1996)
Creators: John Conway
Origin: University of Cambridge, Sidney Sussex College
Country: United Kingdom


FRACTRAN is a Turing-complete esoteric programming language invented by the mathematician John Conway. A FRACTRAN program is an ordered list of positive fractions together with an initial positive integer input n. The program is run by updating the integer n as follows: for the first fraction f in the list for which nf is an integer, replace n by nf repeat this rule until no fraction in the list produces an integer when multiplied by n, then halt. In The Book of Numbers, John Conway and Richard Guy gave a formula for primes in FRACTRAN:                                    (                                       17               91                                   ,                                       78               85                                   ,                                       19               51                                   ,                                       23               38                                   ,                                       29               33                                   ,                                       77               29                                   ,                                       95               23                                   ,                                       77               19                                   ,                                       1               17                                   ,                                       11               13                                   ,                                       13               11                                   ,                                       15               14                                   ,                                       15               2                                   ,                                       55               1                                   )                          {\displaystyle \left({\frac {17}{91}},{\frac {78}{85}},{\frac {19}{51}},{\frac {23}{38}},{\frac {29}{33}},{\frac {77}{29}},{\frac {95}{23}},{\frac {77}{19}},{\frac {1}{17}},{\frac {11}{13}},{\frac {13}{11}},{\frac {15}{14}},{\frac {15}{2}},{\frac {55}{1}}\right)}    Starting with n=2, this FRACTRAN program generates the following sequence of integers: 2, 15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, ... (sequence A007542 in the OEIS) After 2, this sequence contains the following powers of 2:                                    2                        2                             =         4         ,                             2                        3                             =         8         ,                             2                        5                             =         32         ,                             2                        7                             =         128         ,                             2                        11                             =         2048         ,                             2                        13                             =         8192         ,                             2                        17                             =         131072         ,                             2                        19                             =         524288         ,                  …                 {\displaystyle 2^{2}=4,\,2^{3}=8,\,2^{5}=32,\,2^{7}=128,\,2^{11}=2048,\,2^{13}=8192,\,2^{17}=131072,\,2^{19}=524288,\,\dots }    (sequence A034785 in the OEIS) which are the prime powers of 2.