Yellowstone sequence

Yellowstone sequence is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

The Yellowstone sequence, also called the Yellowstone permutation, is defined as:

For n <= 3,

a(n) = n.

For n >= 4,

a(n) = the smallest number not already in sequence such that a(n) is relatively prime to a(n-1) and is not relatively prime to a(n-2).

The sequence is a permutation of the natural numbers, and gets its name from what its authors felt was a spiking, geyser like appearance of a plot of the sequence.

Example

a(4) is 4 because 4 is the smallest number following 1, 2, 3 in the sequence that is relatively prime to the entry before it (3), and is not relatively prime to the number two entries before it (2).

Task

Find and show as output the first 30 Yellowstone numbers.

Extra

Demonstrate how to plot, with x = n and y coordinate a(n), the first 100 Yellowstone numbers.

Related tasks

•   Greatest common divisor.
•   Plot coordinate pairs.

See also

•   The OEIS entry:   A098550 The Yellowstone permutation.
•   Applegate et al, 2015: The Yellowstone Permutation [1].

Factor

Works with: Factor version 0.99 2020-01-23

<lang factor>USING: accessors assocs colors.constants combinators.short-circuit io kernel math prettyprint sequences sets ui ui.gadgets ui.gadgets.charts ui.gadgets.charts.lines ;

yellowstone? ( n hs seq -- ? )

   {
       [ drop in? not ]
       [ nip last gcd nip 1 = ]
       [ nip dup length 2 - swap nth gcd nip 1 > ]
   } 3&& ;

next-yellowstone ( hs seq -- n )

   [ 4 ] 2dip [ 3dup yellowstone? ] [ [ 1 + ] 2dip ] until
   2drop ;

next ( hs seq -- hs' seq' )

   2dup next-yellowstone [ suffix! ] [ pick adjoin ] bi ;

<yellowstone> ( n -- seq )

   [ HS{ 1 2 3 } clone dup V{ } set-like ] dip dup 3 <=
   [ head nip ] [ 3 - [ next ] times nip ] if ;

! Show first 30 Yellowstone numbers.

"First 30 Yellowstone numbers:" print 30 <yellowstone> [ pprint bl ] each nl

! Plot first 100 Yellowstone numbers.

chart new { { 0 100 } { 0 175 } } >>axes line new COLOR: blue >>color 100 <iota> 100 <yellowstone> zip >>data add-gadget "Yellowstone numbers" open-window</lang>

Output:

First 30 Yellowstone numbers:
1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17

Go

This uses Gnuplot-X11 to do the plotting rather than a third party Go plotting library. <lang go>package main

import (

   "fmt"
   "log"
   "os/exec"

)

func gcd(x, y int) int {

   for y != 0 {
       x, y = y, x%y
   }
   return x

}

func yellowstone(n int) []int {

   m := make(map[int]bool)
   a := make([]int, n+1)
   for i := 1; i < 4; i++ {
       a[i] = i
       m[i] = true
   }
   min := 3
   for c := 4; c <= n; c++ {
       for i := min + 1; ; i++ {
           if !m[i] && gcd(a[c-1], i) == 1 && gcd(a[c-2], i) > 1 {
               a[c] = i
               m[i] = true
               if i < min {
                   min = i
               }
               break
           }
       }
   }
   return a[1:]

}

func check(err error) {

   if err != nil {
       log.Fatal(err)
   }

}

func main() {

   x := make([]int, 100)
   for i := 0; i < 100; i++ {
       x[i] = i + 1
   }
   y := yellowstone(100)
   fmt.Println("The first 30 Yellowstone numbers are:")
   fmt.Println(y[:30])
   g := exec.Command("gnuplot", "-persist")
   w, err := g.StdinPipe()
   check(err)
   check(g.Start())
   fmt.Fprintln(w, "unset key; plot '-'")
   for i, xi := range x {
       fmt.Fprintf(w, "%d %d\n", xi, y[i])
   }
   fmt.Fprintln(w, "e")
   w.Close()
   g.Wait()

}</lang>

Output:

The first 30 Yellowstone numbers are:
[1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17]

Julia

<lang julia>using Plots

function yellowstone(N)

   a = [1, 2, 3]
   b = Dict(1 => 1, 2 => 1, 3 => 1)
   while length(a) < N
       for i in 4:typemax(Int)
           if !haskey(b, i) && (gcd(i, a[end]) == 1) && (gcd(i, a[end - 1]) > 1)
               push!(a, i)
               b[i] = 1
               break
           end
       end
   end
   return a

end

println("The first 30 entries of the Yellowstone permutation:\n", yellowstone(30))

x = 1:100 y = yellowstone(100) plot(x, y)

</lang>

Output:

The first 30 entries of the Yellowstone permutation:
[1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 12, 35, 16, 7, 10, 21, 20, 27, 22, 39, 11, 13, 33, 26, 45, 28, 51, 32, 17]

REXX

<lang rexx>/*REXX program calculates any number of terms in the Yellowstone (permutation) sequence.*/ parse arg m . /*obtain optional argument from the CL.*/ if m== | m=="," then m= 30 /*Not specified? Then use the default.*/ !.= 0 /*initialize an array of numbers(used).*/

1. = 0 /*count of Yellowstone numbers in seq. */

$= /*list " " " " " */

     do j=1  until #==m;  prev= # - 1
     if j<5  then do;  #= #+1;   @.#= j;  !.#= j;  !.j= 1;  $= strip($ j);  iterate;  end

        do k=1;   if !.k  then iterate          /*Already used?  Then skip this number.*/
        if gcd(k, @.#)\==1  |  gcd(k, @.prev)<2  then iterate   /*not meet requirement?*/
        #= #+1;   @.#= k;     !.k= 1;   $= $ k  /*bump ctr; assign; mark used; add list*/
        leave                                   /*find the next Yellowstone seq. number*/
        end   /*k*/
     end      /*j*/

say $ /*display a list of a Yellowstone seq. */ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ gcd: parse arg x,y; do until y==0; parse value x//y y with y x; end; return x</lang>

output   when using the default input:

1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17

zkl

Translation of: Julia

This sequence is limited to the max size of a Dictionary, 64k <lang zkl>fcn yellowstoneW{

  Walker.zero().tweak(fcn(a,b){
     foreach i in ([1..]){
        if(not b.holds(i) and i.gcd(a[-1])==1 and i.gcd(a[-2]) >1){

a.del(0).append(i); // only keep last two terms b[i]=True; return(i); }

     }
  }.fp(List(2,3), Dictionary(1,True, 2,True, 3,True))).push(1,2,3);

}</lang> <lang zkl>println("The first 30 entries of the Yellowstone permutation:"); yellowstoneW().walk(30).concat(", ").println();</lang>

Output:

The first 30 entries of the Yellowstone permutation:
1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 12, 35, 16, 7, 10, 21, 20, 27, 22, 39, 11, 13, 33, 26, 45, 28, 51, 32, 17

Plot using Gnuplot <lang zkl>gnuplot:=System.popen("gnuplot","w"); gnuplot.writeln("plot '-'"); yellowstoneW().pump(100,gnuplot.writeln,fcn(n){ String(" ",n) }); gnuplot.writeln("e"); gnuplot.flush(); ask("Hit return to finish"); gnuplot.close();</lang>