# [seqfan] Re: A puzzle from Emissary

Mon Nov 28 17:52:51 CET 2011

```It looks like C=2 and for n>1,
a(n) = 1 if n=2^k a power of 2,
a(n) = 2 otherwise

On Mon, Nov 28, 2011 at 6:04 PM, N. J. A. Sloane <njas at research.att.com> wrote:
> Dear Seq Fans,
> Victor Miller posted the following on the math-fun list:
>
>> In the lastest issue of Emissary (the newsletter from MSRI) there's the
>> following challenging puzzle:
>>
>> For every positive integer n, write it in binary, and allow the following
>> possible set of transformations:
>>
>> You can insert "+" signs at arbitrary points within the binary expansion,
>> and interpret that as a sum of binary numbers.  For example
>>
>> 110101_2  --> 11_2+01_2+01_2 = 3 + 1 + 1 = 5 = 101_2.
>>
>> Show that there is an absolute constant C such that for any positive
>> integer n there is a sequence of at most C such transformations that
>> results in 1.
>>
>> Note: The minimal value of C is a real shocker
>>
>> This made me wonder about the obvious generalization to other bases, where
>> now we ask for a sequence of such transformation that results in a single
>> digit base b number.
>>
>> Victor
>
> Me: So consider the sequence
> a(n) = smallest number of steps needed to reach 1
>
> I think this begins (for n = 1,2,3,...)
> 0,1,2,1,2,2,2,1,2,2,2,2,2,2,2,1,...
>
> For example, 14 = 1110_2 -> 1 + 11 + 0 = 4 = 100_2 -> 1 + 0 + 0 = 1
> so a(14)=2
>
> Could someone check this and extend it?
>
> Neil
>
>
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>

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