A076228 Number of terms k in the trajectory of the Collatz function applied to n such that k < n.

0, 1, 2, 2, 3, 5, 4, 3, 6, 5, 6, 8, 6, 9, 6, 4, 8, 13, 10, 7, 5, 11, 8, 10, 13, 9, 9, 15, 13, 10, 9, 5, 16, 11, 8, 19, 17, 16, 17, 8, 12, 7, 19, 15, 13, 11, 12, 11, 19, 20, 17, 11, 9, 17, 14, 19, 23, 18, 21, 15, 13, 16, 14, 6, 22, 24, 21, 14, 12, 11, 15, 22, 18, 21, 7, 21, 19, 25, 22, 9

Offset: 1

Author

Labos Elemer, Oct 01 2002

Keywords

• nonn

Comments

It is believed that for each x, a(n) = x occurs a finite number of times and the largest n is 2^x.

Original name: Start iteration of Collatz-function (A006370) with initial value of n. a(n) shows how many times during fixed-point-list, the value sinks below initial one until reaching endpoint = 1. - Michael De Vlieger, Dec 13 2018

Examples

			A070165(18) = {18, 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1}. a(18) = 13 because 13 terms are smaller than n = 18; namely: {9, 14, 7, 11, 17, 13, 10, 5, 16, 8, 4, 2, 1}.
		

Links

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

Crossrefs

Cf. A006370, A070165, A074473, A159999.

Programs

Mathematica

f[x_] := (1-Mod[x, 2])*(x/2)+(Mod[x, 2])*(3*x+1) f[1]=1; f0[x_] := Delete[FixedPointList[f, x], -1] f1[x_] := f0[x]-Part[f0[x], 1] f2[x_] := Count[Sign[f1[x]], -1] Table[f2[w], {w, 1, 256}]
(* Second program: *)
Table[Count[NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, n, # > 1 &], ?(# < n &)], {n, 80}] (* _Michael De Vlieger, Dec 09 2018 *)