A221947 Smallest number k (different from a power of 2) such that A006577(n*k) = A006577(n) + A006577(k), or 0 if no such number exists.

3, 3, 423, 3, 81, 423, 75, 3, 0, 81, 11003, 423, 155, 75, 35, 3, 239, 0, 151, 81, 23, 11003, 21, 423, 21, 155, 341, 75, 201, 35, 75, 3, 21, 239, 15, 0, 113, 151, 21, 81, 635, 23, 1131, 11003, 2017, 21, 75, 423, 1267, 21, 75, 155, 253, 341, 151, 75, 7931, 201, 75, 35, 69, 75, 213, 3, 1073, 21, 423, 239, 61, 15

Offset

1,1

Comments

A006577 is the number of halving and tripling steps to reach 1 in the '3x+1' problem. If n is a power of 2, a(n) = 3.

If k is a power of 2, we obtain trivial results, for example A006577(n*2^m) = A006577(2^m) + A006577(n) = m + A006577(n) => the smallest k is 1.

It appears that a(n) = 0 for n of the form 9*2^a = 9, 18, 36, 72, ...

Links

Table of n, a(n) for n=1..70.

Index entries for sequences related to 3x+1 (or Collatz) problem

Example

a(3) = 423 because A006577(3*423) = A006577(1269) = 39, and A006577(3) + A006577(423) = 7 + 32 = 39.

Maple

lst:={ }:C:= proc(n) a := 0 ; x := n ; while x > 1 do if type(x, 'even') then x := x/2:a:=a+1:  else x := 3*x+1 ; a := a+1 ; end if; end do; a ; end proc:
for m from 0 to 40 do:lst:=lst union {2^m}:od:for n from 1 to 73 do: ii:=0:for k from 2 to 50000 while(ii=0) do:z:=n*k : if {k} intersect lst = {} and C(z)=C(n)+C(k) then ii:=1: printf ( "%d %d \n", n, k):else fi:od: if ii=0 and {n} intersect lst = {} and {k} intersect lst = {} then printf ( "%d %d \n", n, 0):else fi:od:

Crossrefs

Cf. A006577, A057716.

Sequence in context: A335258 A292163 A242886 * A138662 A009011 A291095

Adjacent sequences: A221944 A221945 A221946 * A221948 A221949 A221950

Keyword

nonn

Author

Michel Lagneau, Feb 25 2013

Status

approved