A282408 Numbers k for which the number of odd members and the number of even members in the Collatz (3x+1) trajectory are both a perfect square.

1, 2, 16, 17, 322, 323, 512, 1595, 1598, 1599, 1609, 1614, 1615, 1627, 1641, 1643, 1657, 1663, 1776, 1780, 1781, 1784, 1786, 2176, 2208, 2216, 2218, 2240, 2256, 2260, 2261, 2274, 2275, 2400, 2408, 2410, 2416, 2417, 2420, 2421, 3844, 3845, 3846, 3848, 3850, 3852

Offset

1,2

Comments

Or numbers m such that A078719(m) and A006666(m) are both a perfect square.

For k < 5*10^6, the fourteen distinct pairs of squares in the order of appearance are: (1, 0), (1, 1), (1, 4), (4, 9), (36, 64), (1, 9), (25, 49), (4, 16), (16, 36), (9, 25), (1, 16), (81, 144), (4, 25) and (64, 121).

The numbers 2^(m^2) = A002416(m) are in the sequence, and the corresponding pairs of squares are (1, m^2).

Number of terms <= 10^h: 2, 4, 7, 202, 203, 474, 20888, etc. Robert G. Wilson v, Feb 14 2017

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Index entries for sequences related to 3x+1 (or Collatz) problem

Example

17 is in the sequence because the Collatz trajectory is 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 => the number of odd members is 4 = 2^2 and number of even members is 9 = 3^2.

Maple

nn:=10^6:
for n from 1 to 10000 do:
  m:=n:i1:=1:i2:=0:
   for i from 1 to nn while(m<>1) do:
    if irem(m, 2)=0
     then
     m:=m/2:i2:=i2+1:
     else
     m:=3*m+1:i1:=i1+1:
    fi:
   od:
    if sqrt(i1)=floor(sqrt(i1)) and sqrt(i2)=floor(sqrt(i2))
     then
     printf(`%d, `, n):
     else
    fi:
od:

Mathematica

fQ[n_] := Block[{m = n, e = 0, o = 1}, While[m > 1, If[OddQ@ m, m = 3 m + 1; o++, m /= 2; e++]]; IntegerQ@ Sqrt@ e && IntegerQ@ Sqrt@ o]; Select[ Range@ 3855, fQ] (* Robert G. Wilson v, Feb 14 2017 *)
memsqQ[n_]:=Module[{col=NestWhileList[If[EvenQ[#], #/2, 3#+1]&, n, #!=1&]}, AllTrue[ {Sqrt[ Count[ col, _?(EvenQ[#]&)]], Sqrt[Count[col, _?(OddQ[ #]&)]]}, IntegerQ]]; Select[Range[4000], memsqQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 04 2018 *)

Crossrefs

Cf. A000079, A002416, A078719, A006666, A006667.

Sequence in context: A342030 A368404 A217307 * A282838 A180923 A058242

Adjacent sequences: A282405 A282406 A282407 * A282409 A282410 A282411

Keyword

nonn

Author

Michel Lagneau, Feb 14 2017

Status

approved