A225089 a(n) = floor(2^A006666(m)/3^A006667(m)) - m, where m = 2n + 1.

0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 1, 2, 1, 2, 1, 0, 1, 4, 3, 1, 0, 3, 2, 4, 4, 2, 5, 5, 4, 3, 5, 0, 6, 3, 2, 8, 7, 6, 8, 2, 7, 0, 10, 6, 5, 4, 7, 8, 10, 9, 8, 4, 3, 10, 9, 11, 14, 9, 12, 7, 6, 10, 9, 0, 14, 13, 12, 7, 6, 5, 10, 17, 13, 15, 0, 13, 12, 16, 15, 5, 8

Offset: 1

Author

Michel Lagneau, Apr 27 2013

Keywords

• nonn

Comments

A006666 and A006667 are the number of halving and tripling steps to reach 1 in 3x+1 problem.

Properties of this sequence:

a(m) = 0 for m = A211981(m).

Examples

			a(9) = 3 because floor(2^A006666(19)/3^A006667(19)) - 19 = floor(2^14 /3^6) - 19 = floor(22.474622) - 19 = 22 - 19 = 3.
		

Links

• Michel Lagneau, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006666, A006667, A211981, A075680.

Programs

Maple

A:= proc(n) if type(n, 'even') then n/2; else 3*n+1 ; end if; end proc:
B:= proc(n) a := 0 ; x := n ; while x > 1 do x := A(x) ; a := a+1 ; end do; a ; end proc:
C:= proc(n) a := 0 ; x := n ; while x > 1 do if type(x, 'even') then x := x/2 ; else x := 3*x+1 ; a := a+1 ; end if; end do; a ; end proc:
D:= proc(n) C(n) ; end proc:
A006666:= proc(n) B(n)- C(n) ; end:
A006667:= proc(n) C(n)- D(n) ; end:
G:= proc(n) floor(2^A006666 (n)/3^A006667 (n)) ; end:
for i from 1 to 100 do: printf(`%d, `, G(i)-i):od:

Mathematica

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; nn = 100; t = {}; n = 0; While[Length[t] < nn, n++; c = Collatz[n]; ev = Length[Select[c, EvenQ]]; od = Length[c] - ev - 1; AppendTo[t, Floor[2^ev/3^od]-n]]; t