A183133 -id:A183133 - cp's OEIS frontend

A183132 Successive integers produced by Conway's PRIMEGAME using Kilminster's Fractran program with only nine fractions.

10, 5, 36, 858, 234, 5577, 1521, 3549, 8281, 910, 100, 50, 25, 180, 3388, 924, 252, 6006, 1638, 39039, 10647, 24843, 57967, 6370, 700, 300, 7150, 1950, 46475, 12675, 29575, 3250, 360, 6776, 1848, 504, 12012, 3276, 78078, 21294, 507507, 138411, 322959, 753571

Offset: 1

Alois P. Heinz, Dec 26 2010

The exponents of exact powers of 10 in this sequence are 1, followed by the successive primes (A008578).

D. Olivastro, Ancient Puzzles. Bantam Books, NY, 1993, p. 21.

Alois P. Heinz, Table of n, a(n) for n = 1..10547
J. H. Conway, FRACTRAN: a simple universal programming language for arithmetic, in T. M. Cover and Gopinath, eds., Open Problems in Communication and Computation, Springer, NY 1987, pp. 4-26.
Esolang wiki, Fractran
Chaim Goodman-Strauss, Can’t Decide? Undecide!, Notices of the AMS, Volume 57, Number 3, pp. 343-356, March 2010.
R. K. Guy, Conway's prime producing machine, Math. Mag. 56 (1983), no. 1, 26-33.
Eric Weisstein's World of Mathematics, FRACTRAN.
Wikipedia, FRACTRAN.

Cf. A183133, A008578, A007542, A007546, A007547.

l:= [3/11, 847/45, 143/6, 7/3, 10/91, 3/7, 36/325, 1/2, 36/5]:
a:= proc(n) option remember;
      global l;
      local p, k;
      if n=1 then 10
             else p:= a(n-1);
                  for k while not type(p*l[k], integer)
                  do od; p*l[k]
      fi
    end:
seq(a(n), n=1..50);

l = {3/11, 847/45, 143/6, 7/3, 10/91, 3/7, 36/325, 1/2, 36/5};
a[n_] := a[n] = Module[{p, k}, If[n == 1, 10, p = a[n - 1]; For[k = 1, !IntegerQ[p*l[[k]]], k++]; p*l[[k]]]];
Array[a, 50] (* Jean-François Alcover, May 28 2018, from Maple *)

from fractions import Fraction
nums = [ 3, 847, 143, 7, 10, 3,  36, 1, 36]
dens = [11,  45,   6, 3, 91, 7, 325, 2,  5]
PRIMEGAME = [Fraction(num, den) for num, den in zip(nums, dens)]
def succ(n, program):
    for i in range(len(program)):
      if (n*program[i]).denominator == 1: return (n*program[i]).numerator
def orbit(start, program, steps):
    orb = [start]
    for s in range(1, steps): orb.append(succ(orb[-1], program))
    return orb
print(orbit(10, PRIMEGAME, steps=44)) # Michael S. Branicky, Oct 05 2021

A267572 Number of steps J. H. Conway's Fractran program needs to calculate the n-th prime.

Original entry on oeis.org

19, 50, 211, 577, 2083, 3469, 7361, 10395, 17915, 35249, 43188, 72392, 97236, 113324, 146556, 209098, 285307, 317925, 417234, 494939, 541264, 684114, 789130, 968524, 1249354, 1408123, 1500944, 1679217, 1781388, 1980305

Offset: 1

Ivan N. Ianakiev, Jan 17 2016

nonn

The sieve consists of the fractions {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}.

			For n = 1, start with 2^n and find the first fraction (fraction1 = 15/2) where the product (2^n)*fraction1 is an integer (integer1 = 15). With integer1 repeat the above, i.e., find the first fraction (fraction2 = 55/1) where integer1*fraction2 is an integer (integer2 = 825). Repeat until a power of 2 is reached (2^2 in this case). The exponent of 2 is prime(1) and a(1) = 19 is the number of steps to reach it.

Dominic Olivastro, Ancient Puzzles, Bantam Books, 1993, pp. 20-21.

J. H. Conway, FRACTRAN: a simple universal programming language for arithmetic, in T. M. Cover and Gopinath, eds., Open Problems in Communication and Computation, Springer, NY 1987, pp. 4-26.
Esolang Wiki "Fractran".
R. K. Guy, Conway's prime producing machine, Math. Mag. 56 (1983), no. 1, 26-33.
Eric Weisstein's World of Mathematics, FRACTRAN

Cf. A007542, A007546, A007547, A183132, A183133.

fracList = {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};
stepCount[n_] := n * fracList[[First[Flatten[Position[n * fracList, First[Select[n * fracList, IntegerQ]]]]]]];
A267572[n_] := Length[NestWhileList[stepCount[#] &, 2^n, stepCount[#] != 2^Prime[n] &]];
Table[tempVar = A267572[n]; Print["a(", n,") = ", tempVar]; tempVar, {n, 30}]

A334722 Numerators of fractions in Kilminster's FracTran program for prime numbers, 10-fraction version.

Original entry on oeis.org

7, 99, 13, 39, 36, 10, 49, 7, 1, 91

Offset: 1

Alonso del Arte, May 09 2020

Like Conway's PRIMEGAME (A202138/A203363), Kilminster's program delivers the prime numbers as exponents of powers of a base, but the base is 10 rather than 2.

John H. Conway and Tim Hsu, Some Very Interesting Sequences, in T. Shubin, D. F. Hayes, and G. Alexanderson (eds.), Expeditions in Mathematics, MAA Spectrum series, Washington, DC, 2011, chapter 6, pp. 75-86. See page 78.

John H. Conway and Tim Hsu, Some Very Interesting Sequences, 2006, p. 4.

Cf. A334723, A202138, A203363.

Kilminster also came up with a 9-fraction program. See A183132, A183133.

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A183132 Successive integers produced by Conway's PRIMEGAME using Kilminster's Fractran program with only nine fractions.

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A267572 Number of steps J. H. Conway's Fractran program needs to calculate the n-th prime.

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A334722 Numerators of fractions in Kilminster's FracTran program for prime numbers, 10-fraction version.

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