A174539 Starting numbers n such that the number of halving and tripling steps to reach 1 under the Collatz 3x+1 map is a perfect square.

1, 2, 7, 12, 13, 16, 44, 45, 46, 80, 84, 85, 98, 99, 100, 101, 102, 107, 129, 153, 156, 157, 158, 169, 272, 276, 277, 280, 282, 300, 301, 302, 350, 351, 512, 576, 592, 596, 597, 608, 616, 618, 625, 642, 643, 644, 645, 646, 648, 649, 650, 651, 652, 653, 654, 655, 662, 663

Offset: 1

Author

Michel Lagneau, Mar 21 2010

Keywords

• nonn

Comments

Numbers n such that A006577(n) is a perfect square.

Examples

			44, 45 and 46 are in the sequence because the number of steps as counted in A006577 for each of them is 16 = 4^2, a perfect square.
		

Links

• Index to sequences related to the 3x+1 problem

Crossrefs

Cf. A006577, A008908, A064433, A006666, A006667, A033491

Programs

Maple

with(numtheory):for x from 1 to 200 do traj:=0: n1:=x: x1:=x: for p from 1 to 20 while(irem(x1,2)=0)do p1:=2^p: xx1:=x1: x1:=floor(n1/p1): traj:=traj+1:od:
n:=x1: for q from 1 to 100 while(n<>1)do n1:=3*n+1: traj:=traj+1: x0:=irem(n1,2): for p from 1 to 20 while(x0=0)do p1:=2^p: xx1:=x1: x1:=floor(n1/p1): x0:=n1-p1*x1: traj:=traj+1: od: traj:=traj-1: n:=xx1:od:
if(sqrt(traj))=floor(sqrt(traj)) then print(x):else fi:od:

Mathematica

htsQ[n_]:=With[{len=Length[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#!=1&]]-1},IntegerQ[Sqrt[len]]]; Select[Range[700],htsQ] (* Harvey P. Dale, Jan 01 2023 *)

Formula

{n: A006577(n) in A000290}.

Extensions

Unspecific references removed - R. J. Mathar, Mar 31 2010

Corrected and extended by Harvey P. Dale, Jan 01 2023