[seqfan] Re: Colorless integers

Jack Brennen jfb at brennen.net
Tue Feb 22 22:39:41 CET 2011

```Cycles must consist exclusively of multiples of 9.

Any time that a number's successor is smaller than the first number, the
successor will be a multiple of 9.  Any multiple of 9 has a successor
which is a multiple of 9.  Every cycle will have a downward step in the
cycle.  Put all that together, it's easy to see that all cycles are
entirely multiples of 9.

On 2/22/2011 1:01 PM, William Keith wrote:
> On Tue, Feb 22, 2011 at 7:56 PM, N. J. A. Sloane<njas at research.att.com>wrote:
>
>> Start at n, let a(n) be the smallest number in the cycle if it
>> goes into a cycle, or -1 if it diverges. Is this sequence
>> in the OEIS? and if not, you know what to do!
>> Neil
>>
>
> Looks like none of them diverge, none go to 0 after 98, and all of them wind
> up in a cycle that is fairly close by.  Cycles appear to be exclusively
> rooted on multiples of 9: {99, 117, 108, 99}, {198, 216, 207, 198}, {6732,
> 6750, 6768, 6741, 6759, 6732}, {6822, 6840, 6858, 6831, 6849, 6822}, etc.
> Makes sense, as sums of digits will be closed under that property, and I'd
> suspect that there's a digital criterion that will tell you where a number
> goes.
>
> William Keith
>
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```