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

1,2

Comments

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

Links

Table of n, a(n) for n=1..58.

Index to sequences related to the 3x+1 problem

Formula

{n: A006577(n) in A000290}.

Example

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.

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 *)

Crossrefs

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

Sequence in context: A226703 A126343 A344951 * A049409 A287580 A326231

Adjacent sequences: A174536 A174537 A174538 * A174540 A174541 A174542

Keyword

nonn

Author

Michel Lagneau, Mar 21 2010

Extensions

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

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

Status

approved