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

Author

Michel Lagneau, Feb 14 2017

Keywords

• nonn

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

Examples

			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.
		

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

Crossrefs

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

Programs

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 *)