Fractran

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FRACTRAN is a Turing-complete esoteric programming language invented by the mathematician John Horton Conway.

A FRACTRAN program is an ordered list of positive fractions $P = (f_1, f_2, \ldots, f_m)$, together with an initial positive integer input $n$.

The program is run by updating the integer $n$ as follows:

• for the first fraction, $f_i$, in the list for which $nf_i$ is an integer, replace $n$ with $nf_i$ ;
• repeat this rule until no fraction in the list produces an integer when multiplied by $n$, then halt.

Conway gave a program for primes in FRACTRAN:

$\dfrac{17}{91}$, $\dfrac{78}{85}$, $\dfrac{19}{51}$, $\dfrac{23}{38}$, $\dfrac{29}{33}$, $\dfrac{77}{29}$, $\dfrac{95}{23}$, $\dfrac{77}{19}$, $\dfrac{1}{17}$, $\dfrac{11}{13}$, $\dfrac{13}{11}$, $\dfrac{15}{14}$, $\dfrac{15}{2}$, $\dfrac{55}{1}$

Starting with $n=2$, this FRACTRAN program will change $n$ to $15=2\times (\frac{15}{2})$, then $825=15\times (\frac{55}{1})$, generating the following sequence of integers:

$2$, $15$, $825$, $725$, $1925$, $2275$, $425$, $390$, $330$, $290$, $770$, $\ldots$

After 2, this sequence contains the following powers of 2:

$2^2=4$, $2^3=8$, $2^5=32$, $2^7=128$, $2^{11}=2048$, $2^{13}=8192$, $2^{17}=131072$, $2^{19}=524288$, $\ldots$

which are the prime powers of 2.