A202138 -id:A202138 - cp's OEIS frontend

A007542 Successive integers produced by Conway's PRIMEGAME.

2, 15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, 910, 170, 156, 132, 116, 308, 364, 68, 4, 30, 225, 12375, 10875, 28875, 25375, 67375, 79625, 14875, 13650, 2550, 2340, 1980, 1740, 4620, 4060, 10780, 12740, 2380, 2184, 408, 152

Offset: 1

N. J. A. Sloane

Conway's PRIMEGAME produces the terms 2^prime in increasing order.
From Daniel Forgues, Jan 20 2016: (Start)
Pairs (n, a(n)) such that a(n) = 2^k are (1, 2^1), (20, 2^2), (70, 2^3), (282, 2^5), (711, 2^7), (2376, 2^11), (3894, 2^13), (8103, 2^17), ...
Numbers n such that a(n) = 2^k are 1, 20, 70, 282, 711, 2376, 3894, 8103, ... [This is 1 + A007547. - N. J. A. Sloane, Jan 25 2016] (End)

D. Olivastro, Ancient Puzzles. Bantam Books, NY, 1993, p. 21.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n=1..8103
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.
Richard K. Guy, Conway's prime producing machine, Math. Mag. 56 (1983), no. 1, 26-33. [Gives slightly different version of the program which produces different terms, starting from n=132.]
OEIS Wiki, Conway's PRIMEGAME
Kevin I. Piterman and Leandro Vendramin, Computer algebra with GAP, 2023. See p. 40.
Leandro Vendramin, Mini-couse on GAP - Exercises, Universidad de Buenos Aires (Argentina, 2020).
Eric Weisstein's World of Mathematics, FRACTRAN
Wikipedia, FRACTRAN

Cf. A007546, A007547, A183132, A202138, A203363.

a007542 n = a007542_list !! (n-1)
a007542_list = iterate a203907 2  -- Reinhard Zumkeller, Jan 24 2012

l:= [17/91, 78/85, 19/51, 23/38, 29/33, 77/29, 95/23, 77/19, 1/17, 11/13, 13/11, 15/2, 1/7, 55]: a:= proc(n) option remember; global l; local p, k; if n=1 then 2 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); # Alois P. Heinz, Aug 12 2009

conwayFracs := {17/91, 78/85, 19/51, 23/38, 29/33, 77/29, 95/23, 77/19, 1/17, 11/13, 13/11, 15/2, 1/7, 55}; a[1] = 2; A007542[n_] := A007542[n] = (p = A007542[n - 1]; k = 1; While[ ! IntegerQ[p * conwayFracs[[k]]], k++]; p * conwayFracs[[k]]); Table[A007542[n], {n, 42}] (* Jean-François Alcover, Jan 23 2012, after Alois P. Heinz *)

from fractions import Fraction
nums = [17, 78, 19, 23, 29, 77, 95, 77,  1, 11, 13, 15, 1, 55] # A202138
dens = [91, 85, 51, 38, 33, 29, 23, 19, 17, 13, 11,  2, 7,  1] # A203363
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(2, PRIMEGAME, steps=42)) # Michael S. Branicky, Feb 15 2021

a(n+1) = A203907(a(n)), a(1) = 2. [Reinhard Zumkeller, Jan 24 2012]

A203363 Denominators of Conway's PRIMEGAME.

Original entry on oeis.org

91, 85, 51, 38, 33, 29, 23, 19, 17, 13, 11, 2, 7, 1

Offset: 1

Alonso del Arte, Dec 31 2011

Numerators are in A202138.

Notice that the last number of this sequence is 1, meaning that the last number in the "prime producing machine" is an integer, namely 55. Thus, if each multiplication by the previous numbers of the "machine" fails to give an integer, this one will, and it will be a multiple of 55.

Cf. A007542

a203363_list = [91, 85, 51, 38, 33, 29, 23, 19, 17, 13, 11, 2, 7, 1]
-- Reinhard Zumkeller, Jan 24 2012

A203907 Successor function for Conway's PRIMEGAME.

Original entry on oeis.org

55, 15, 165, 30, 275, 45, 1, 60, 495, 75, 13, 90, 11, 105, 825, 120, 1, 135, 77, 150, 3, 26, 95, 180, 1375, 22, 1485, 210, 77, 225, 1705, 240, 29, 2, 5, 270, 2035, 23, 33, 300, 2255, 315, 2365, 52, 2475, 190, 2585, 360, 7, 375, 19, 44, 2915, 405, 65, 420

Offset: 1

Reinhard Zumkeller, Jan 24 2012

a(n) <= 55 * n, as 55/1 is the last and largest FRACTRAN fraction.
Iterations, starting with 2, give A007542. A185242 begins with 3.
A quasipolynomial of order 6469693230 = 29#. - Charles R Greathouse IV, Jul 31 2016
Apparent simple regularities do not necessarily hold. It is true that a(2n)/15 = a(4n)/30, but for n = 11, 13, 17, 19, 22, 23, ... this is not equal to n. Also, a(2k-1) = 55k holds for more than 60%, but not for all k >= 1. - M. F. Hasler, Jun 15 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, FRACTRAN
Wikipedia, Conway's PRIMEGAME

Cf. A007542.

import Data.Ratio ((%), numerator, denominator)
a203907 n = numerator $ head
   [x | x <- map (* fromInteger n) fracts, denominator x == 1]
   where fracts = zipWith (%) a202138_list a203363_list
a203907_list = map a203907 [1..]

conwayFracs = {17/91, 78/85, 19/51, 23/38, 29/33, 77/29, 95/23, 77/19, 1/17, 11/13, 13/11, 15/2, 1/7, 55}; conwayProc[n_] := Module[{curr = 1/2, iter = 1}, While[Not[IntegerQ[curr]], curr = conwayFracs[[iter]]n; iter++]; Return[curr]]; Table[conwayProc[n], {n, 60}] (* Alonso del Arte, Jan 24 2012 *)

{A203907(n,V=[17/91, 78/85, 19/51, 23/38, 29/33, 77/29, 95/23, 77/19, 1/17, 11/13, 13/11, 15/2, 1/7, 55])=for(i=1,#V, denominator(V[i]*n)==1 && return(V[i]*n))} \\ Charles R Greathouse IV, Jul 31 2016, edited by M. F. Hasler, Jun 15 2017

Let [17/91, 78/85, 19/51, 23/38, 29/33, 77/29, 95/23, 77/19, 1/17, 11/13, 13/11, 15/2, 1/7, 55/1] be the list of FRACTRAN fractions = [A202138(k)/A203363(k) : 1<=k<=14], then a(n) = n*f, where f is the first term yielding an integral product.

A275483 Numerators of Conway's FIBONACCIGAME.

Original entry on oeis.org

17, 133, 17, 23, 2233, 23, 31, 74, 31, 41, 129, 41, 13, 1, 1

Offset: 1

Alonso del Arte, Jul 30 2016

Like PRIMEGAME, the object of FIBONACCIGAME is to come up with a listing of the Fibonacci numbers through a process of multiplying integers by fractions to see which produce integers, which are then cycled back through the process.

However, unlike PRIMEGAME, FIBONACCIGAME starts from 78 * 5^(n - 1) and stops at 2^Fibonacci(n).

Julian Havil, Nonplussed! Mathematical Proof of Implausible Ideas. Princeton: Princeton University Press (2007): 174.

Cf. A275484 (denominators), A202138, A000301.

A350555 Numerators of Conway's PIGAME.

Original entry on oeis.org

365, 29, 79, 679, 3159, 83, 473, 638, 434, 89, 17, 79, 31, 41, 517, 111, 305, 23, 73, 61, 37, 19, 89, 41, 833, 53, 86, 13, 23, 67, 71, 83, 475, 59, 41, 1, 1, 1, 1, 89

Offset: 1

Paolo Xausa, Jan 05 2022

These rational numbers represent a FRACTRAN program that generates the decimal expansion of Pi (A000796).
Conway proves that, when this program is started at 2^k (with k >= 0), the next power of 2 to appear is 2^Pi_d(k), where Pi_d(0) = 3 and, for k >= 1, Pi_d(k) is the k-th digit after the point in the decimal expansion of Pi.
According to Kaushik, Murphy, and Weed, the starting value should be 89*2^k. - Andrei Zabolotskii, Aug 23 2025

J. H. Conway, "FRACTRAN: A Simple Universal Programming Language for Arithmetic", in J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, American Mathematical Society, 2010, p. 249, and in T. M. Cover and B. Gopinath, eds, Open Problems in Communication and Computation, Springer, New York, NY, 1987, pp. 4-26.
Khushi Kaushik, Tommy Murphy, and David Weed, Computing sqrt(2) with FRACTRAN, Irish Math. Soc. Bull., 95 (2025), 23-34 (warning: theorem 3.1 is missing the last fraction 1/97); arXiv:2412.16185 [cs.PL], 2024.
Wikipedia, FRACTRAN.

Cf. A000796, A202138, A350556 (denominators).

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.

A350663 Numerators of Conway's POLYGAME.

Original entry on oeis.org

583, 629, 437, 82, 615, 371, 1, 53, 43, 23, 341, 41, 47, 29, 37, 37, 299, 47, 161, 527, 159, 1, 1, 1

Offset: 1

Paolo Xausa, Jan 10 2022

These rational numbers represent a FRACTRAN program capable of calculating any computable function.
If, when started at c*2^(2^k), the program stops at 2^(2^m), then c encodes the computable function f_c, and f_c(k) = m, where c, k and m are nonnegative integers.
In the linked work Conway lists some values of c (which he calls "catalog numbers") encoding various simple functions, including the (extremely large) value of c for computing the k-th digit in the decimal expansion of Pi.

J. H. Conway, "FRACTRAN: A Simple Universal Programming Language for Arithmetic", in T. M. Cover and B. Gopinath, eds, Open Problems in Communication and Computation, Springer, New York, NY, 1987, pp. 4-26.
J. H. Conway, "FRACTRAN: A Simple Universal Programming Language for Arithmetic", in J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, American Mathematical Society, 2010, pp. 249-264.
Wikipedia, FRACTRAN.

Cf. A202138, A350555, A350664 (denominators).

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A007542 Successive integers produced by Conway's PRIMEGAME.

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A203363 Denominators of Conway's PRIMEGAME.

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A203907 Successor function for Conway's PRIMEGAME.

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A275483 Numerators of Conway's FIBONACCIGAME.

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A350555 Numerators of Conway's PIGAME.

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

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A350663 Numerators of Conway's POLYGAME.

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