A266796 Numbers n such that the concave polygon formed by the trajectory of n in the "3n+1" problem and the straight line between the coordinate points (0,n) and (r,1) where r is the number of iterations needed to reach 1 is not a self-intersecting polygon.

3, 4, 5, 7, 8, 11, 16, 27, 31, 32, 40, 47, 48, 63, 64, 71, 80, 88, 96, 104, 112, 116, 128, 136, 144, 148, 152, 160, 168, 176, 180, 192, 196, 200, 208, 212, 224, 232, 256, 260, 264, 272, 276, 280, 288, 296, 304, 308, 320, 336, 344, 352, 360, 368, 372, 384

Offset

1,1

Comments

The finite sequence A265687 is included in this sequence and considers only the polygons located above the line connecting (0,n) and (r,1). This sequence considers all polygons located either above or below the line connecting (0,n) and (r,1).

Consider the 3n+1 function iterates (n, T(n), T(T(n)),...,4,2,1) plotted on standard vertical and horizontal scales. Each coordinate point (0,n), (1,T(n)), (2, T(T(n)),...,(r,1), where r is the number of iterations needed to reach 1, is connected to the next by a straight line, and the two points (0,n) and (r,1) are also connected by a straight line in order to form a polygon. The sequence lists the numbers n such that all the points of the polygon are located either above the line connecting (0,n) and (r,1) or below this line.

Properties of the sequence:

The powers of 2 (A000079) are in the sequence, except the numbers 1 and 2.

The odd values of the sequence are 3, 5, 7, 11, 27, 31, 47, 63 and 71.

Links

Table of n, a(n) for n=1..56.

Michel Lagneau, Examples

Example

see the link.

Maple

T:=array(1..2000):U:=array(1..2000):nn:=100:
for n from 2 to nn do:
  kk:=1:m:=n:T[kk]:=n:it:=0:
    for i from 1 to nn while(m<>1) do:
     if irem(m, 2)=0
      then
       m:=m/2:kk:=kk+1:T[kk]:=m:it:=it+1:
      else
      m:=3*m+1:kk:=kk+1:T[kk]:=m:it:=it+1:
     fi:
    od:
    for j from 1 to it do:
     z:=((1-n)/it)*(j-1)+n:
      if z<T[j] then U[j]:=1
      else
      U[j]:=-1
      fi:
     od:
       jj:=0:
       for k from 2 to it-1 do:
        if U[k]+U[k+1]=0 then jj:=jj+1:
         else fi:
       od:
         if jj=0
         then
         printf(`%d, `, n):
         else fi:
       od:

Crossrefs

Cf. A006577, A265687.

Sequence in context: A268678 A057201 A242965 * A318603 A154571 A174269

Adjacent sequences: A266793 A266794 A266795 * A266797 A266798 A266799

Keyword

nonn

Author

Michel Lagneau, Jan 03 2016

Status

approved