A203363 -id:A203363 - 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]

A202138 Numerators of Conway's PRIMEGAME.

Original entry on oeis.org

17, 78, 19, 23, 29, 77, 95, 77, 1, 11, 13, 15, 1, 55

Offset: 1

Alonso del Arte, Dec 31 2011

Denominators are in A203363.
Conway's PRIMEGAME (also called "Conway's prime producing machine") is a fascinating (and very inefficient) method for obtaining the prime numbers.
The "machine" consists of 14 rational numbers. Starting with 2, one finds the first number in the machine that multiplied by 2 gives an integer; then for that integer we find the first number in the machine that generates another integer. This process is repeated for each new integer obtained. Thus A007542 is generated. Except for the initial 2, each number in A007542 having an integer for a binary logarithm is a prime number.
Note that in R. K. Guy's 1983 paper, the last four numbers of the machine are 13/11, 15/14, 15/2 and 55 rather than 13/11, 15/2, 1/7 and 55.

R. K. Guy, Conway's prime producing machine, Math. Mag. 56 (1983), no. 1, 26-33.

Cf. A007546, A007547.

a202138_list = [17, 78, 19, 23, 29, 77, 95, 77, 1, 11, 13, 15, 1, 55]
-- 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.

A275484 Denominators of Conway's FIBONACCIGAME.

Original entry on oeis.org

65, 34, 19, 17, 69, 29, 23, 341, 37, 31, 287, 43, 41, 13, 3

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.

Notice that there isn't a 1 in this sequence, which means that the progress of the program, starting from a valid input, eventually halts with 2^Fibonacci(n).

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

Cf. A275483 (numerators), A203363, A000301.

A350556 Denominators of Conway's PIGAME.

Original entry on oeis.org

46, 161, 575, 451, 413, 407, 371, 355, 335, 235, 209, 122, 183, 115, 89, 83, 79, 73, 71, 67, 61, 59, 57, 53, 47, 43, 41, 38, 37, 31, 29, 19, 17, 13, 291, 7, 11, 1024, 97, 1

Offset: 1

Paolo Xausa, Jan 05 2022

See A350555 for numerators, comments and links.

Cf. A000796, A203363, A350555.

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.

A334723 Denominators of fractions in Kilminster's FracTran program for prime numbers, 10-fraction version.

Original entry on oeis.org

3, 98, 49, 35, 91, 143, 13, 11, 2, 1

Offset: 1

Alonso del Arte, May 09 2020

Numerators are in A334722.

Just like the denominators for Conway's PRIMEGAME (A203363), the last denominator in Kilminster's program is 1, meaning the last fraction is an integer. Then, like Conway's PRIMEGAME and unlike Conway's FIBONACCIGAME, Kilminster's program has no halting numbers.

Cf. A334722, A203363.

A350664 Denominators of Conway's POLYGAME.

Original entry on oeis.org

559, 551, 527, 517, 329, 129, 115, 86, 53, 47, 46, 43, 41, 37, 31, 31, 29, 23, 15, 19, 7, 17, 13, 3

Offset: 1

Paolo Xausa, Jan 10 2022

See A350663 for numerators, comments and links.

Cf. A203363, A350556, A350663.

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A007542 Successive integers produced by Conway's PRIMEGAME.

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A202138 Numerators of Conway's PRIMEGAME.

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A203907 Successor function for Conway's PRIMEGAME.

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A275484 Denominators of Conway's FIBONACCIGAME.

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A350556 Denominators 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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A334723 Denominators of fractions in Kilminster's FracTran program for prime numbers, 10-fraction version.

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A350664 Denominators of Conway's POLYGAME.

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