A159999 Number of numbers not greater than n occurring in Collatz (3x+1) trajectory starting with n.

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

Offset: 1

Author

Reinhard Zumkeller, May 04 2009

Keywords

• nonn
• look

Comments

If the Collatz conjecture is true, there are no cycles in the 3x+1 trajectory and the difference between the counts here and those of A076228 is that the start value is counted here but not there; then a(n) = 1+A076228(n) [discovered by sequencedb.net]. - R. J. Mathar, Jun 24 2021

Examples

			a(9) = #{1,2,4,5,7,8,9} = 7, as
9-28-14-7-22-11-34-17-52-26-13-40-20-10-5-16-8-[4-2-1]*
9-..-..-7-..-..-..-..-..-..-..-..-..-..-5-..-8-[4-2-1]*.
		

Links

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

Crossrefs

Cf. A033496, A008908, A070165, A214614.

Programs

Haskell

a159999 n = length $ takeWhile (<= n) $ sort $ a070165_row n
-- Reinhard Zumkeller, Sep 01 2012

Mathematica

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; f[n_] := Module[{c = Collatz[n]}, Length[Select[c, # <= n &]]]; Table[ f[n], {n, 100}] (* T. D. Noe, Mar 07 2013 *)

Formula

a(n) < n for n>6;

a(A033496(n)) = A008908(A033496(n)).

a(n) = f(n,n,1) with f(n,m,x) = if m=1 then x else f(n, A006370(m), if A006370(m)

a(n) = n - A246436(n); row lengths of triangle A214614. - Reinhard Zumkeller, Sep 01 2014

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