A203907 Successor function for Conway's PRIMEGAME.

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

Author

Reinhard Zumkeller, Jan 24 2012

Keywords

• nonn
• easy
• nice

Comments

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

Links

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

Crossrefs

Cf. A007542.

Programs

Haskell

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..]

Mathematica

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 *)

PARI

{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

Formula

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.