A265687 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, 5, 7, 11, 27, 31, 47, 63, 71

Offset

1,1

Comments

This sequence is included in A266796 and considers only the polygons located above 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 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 above the line connecting (0,n) and (r,1).

In other words, the sequence lists the numbers n such that the concave polygons are entirely contained in a half-plane defined by the line passing through the points (0,n) and (r,1), of equation f(x)= x*(1-n)/r + n. We must have f(0)= n, f(1)<T(n), f(2)<T(T(n)),...,f(r-1)<2, f(r)=1.

Conjecture: the sequence is finite.

From David Consiglio, Jr., Jan 07 2016: (Start)

Any additional terms must be congruent to either 3 or 7 mod 8 and larger than 10^8.

Even terms are reduced in just one step, and so immediately fall below the line for any initial number > 2.

Odd terms that are 1 mod 8 or 5 mod 8 reduce to just over 3/4ths their original size in 3 steps. This eliminates any term of this type that requires more than 12 steps from consideration (since 9/12 = 3/4). Since the last number which requires 12 steps or fewer to reduce is 341, and all numbers up to that have been checked, all numbers 1 mod 8 cannot be members of this sequence.

It may be able to further disqualify groups of numbers using this modulus approach.

Links

Table of n, a(n) for n=1..9.

Michel Lagneau, Examples

Example

See the links.

Maple

T:=array(1..1000):nn:=1000:
for n from 3 to 10000 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:
   jj:=0:
    for j from 1 to it+1 do:
     z:=((1-n)/it)*(j-1)+n:
      if z>T[j] then jj:=1:
      else
      fi:
    od:
      if jj=0
       then
       printf(`%d, `, n):
       else
      fi:
od:

Crossrefs

Cf. A006577, A266796.

Sequence in context: A303705 A309520 A344411 * A023368 A295973 A142247

Adjacent sequences: A265684 A265685 A265686 * A265688 A265689 A265690

Keyword

nonn,more

Author

Michel Lagneau, Dec 20 2015

Status

approved