A183132 -id:A183132 - 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]

A183133 Number of steps to compute the n-th prime in PRIMEGAME using Kilminster's Fractran program with only nine fractions.

Original entry on oeis.org

10, 46, 196, 500, 1428, 2488, 4588, 6840, 10546, 17118, 23064, 33332, 44472, 55848, 70330, 90836, 115136, 137912, 168802, 201000, 233542, 276680, 320332, 373198, 439722, 503810, 568334, 640092, 712314, 792186, 917090, 1023878, 1146632, 1263818, 1419298

Offset: 1

Alois P. Heinz, Dec 26 2010

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

Alois P. Heinz, Table of n, a(n) for n = 1..100
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. A183132, A008578, A007542, A007546, A007547.

a:= proc(n) option remember;
      local l,p,m,k;
      l:= [3/11, 847/45, 143/6, 7/3, 10/91, 3/7, 36/325, 1/2, 36/5]:
      if n=1 then b(0):= 10; a(0):= 0
             else a(n-1)
      fi;
      p:= b(n-1);
      for m do
         for k while not type(p*l[k], integer)
         do od; p:= p*l[k];
         if 10^ilog10(p)=p then break fi
      od:
      b(n):= p;
      m + a(n-1)
    end:
seq(a(n), n=1..20);

a[n_] := a[n] = Module[{l, p, m, k},
     l = {3/11, 847/45, 143/6, 7/3, 10/91, 3/7, 36/325, 1/2, 36/5};
     If[n == 1, b[0] = 10; a[0] = 0, a[n - 1]]; p = b[n - 1];
     For[m = 1, True, m++,
          For[k = 1, !IntegerQ[p*l[[k]]], k++];
          p = p*l[[k]];If[10^(Length@IntegerDigits[p]-1) == p, Break[]]];
     b[n] = p; m + a[n - 1]];
Array[a, 20] (* Jean-François Alcover, Apr 02 2021, after Alois P. Heinz *)

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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A007542 Successive integers produced by Conway's PRIMEGAME.

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A183133 Number of steps to compute the n-th prime in 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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