A316783 Seed values of hailstone sequences for which, by the time they first reach 1, the ratio of odd terms to even terms is exactly 1/2. (The seed value itself is included in the ratio calculation.)

4, 5, 6, 11, 14, 15, 18, 19, 25, 33, 43, 57, 59, 78, 79, 105, 135, 139, 185, 187, 191, 246, 247, 249, 254, 255, 329, 338, 339, 359, 427, 438, 439, 443, 450, 451, 478, 479, 569, 585, 590, 591, 601, 636, 637, 638, 758, 759, 767, 779, 786, 787, 801, 849, 850, 851

Offset: 1

Author

Matt Enlow, Jul 13 2018

Keywords

• nonn

Comments

1/2 is the most common odd/even ratio for these sequences. For seed values from 2 to 2^17, it is the odd/even ratio for 4454 of those sequences. The next most common ratio is 5/11, with 1834 sequences. (This is all empirical observation, using Mathematica.)

Examples

			The hailstone sequence starting with 11 (and ending at 1) is 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. Five of those terms are odd, and the other ten are even, giving an odd/even ratio of 1/2. Therefore 11 is a term in this sequence.
		

Links

• Matt Enlow, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A006370, A006666, A078719.

Programs

Maple

b:= proc(n) option remember; `if`(n=1, [1, 0],
      `if`(n::odd, [1, 0]+b(3*n+1), [0, 1]+b(n/2)))
    end:
a:= proc(n) option remember; local k; for k from a(n-1)
      +1 while (l-> 2*l[1]<>l[2])(b(k)) do od; k
    end: a(0):=0:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 02 2018

Mathematica

collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1]&, n, # != 1 &];
A316783 = Select[Range[2, 1000], With[{c = Mod[collatz[#], 2]}, 2 Total[c] == Total[1 - c]] &]

PARI

is(n) = my(x=n, even=0, odd=0); while(1, if(x%2==0, x=x/2; even++, x=3*x+1; odd++); if(x==1, odd++; break)); odd/even==1/2 \\ Felix Fröhlich, Jul 13 2018

Formula

{ k : A006666(k)/A078719(k) = 2 }. - Alois P. Heinz, Aug 02 2018