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

Author

Michel Lagneau, Feb 25 2013

Keywords

• nonn

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, ...

Examples

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

Links

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

Crossrefs

Cf. A006577, A057716.

Programs

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: