# [seqfan] Re: A puzzle from Emissary

Aai agroeneveld400 at gmail.com
Wed Nov 30 15:36:08 CET 2011

```If I'm not wrong then the sequence can be obtained from A000120

1  1 2  1 2 2 3  1 2 2 3 2 3 3 4  1 2 2 3 2 3 3 4 2 3 3 4 3 4 4 5

1  1 2  1 2 2 2  1 2 2 2 2 2 2 2  1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Hallo N. J. A. Sloane, je schreef op 28-11-11 17:04:
> 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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--
Met vriendelijke groet,
@@i=Arie Groeneveld

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