# [seqfan] Re: Fractions, digits, decimal expansion

Bob Selcoe rselcoe at entouchonline.net
Tue Dec 15 03:37:32 CET 2015

```Hi Neil,

I thought my reply (clarification) went through because Eric replied to it -
didn't realize it was off-list Just to him.

Anyway, here's what I said:

Hi Eric and Seqfans;

I may have spoken too soon:  80190/9 = 8910.  It may be impossible for S to
equal 80190 or another "precursor" of 9 but I'm not so sure.

Perhaps someone who knows how to program could expand the sequence and see
if 9 appears?  Can someone offer a proof one way or the other?

Cheers,
Bob

--------------------------------------------------
From: "Neil Sloane" <njasloane at gmail.com>
Sent: Monday, December 14, 2015 7:53 PM
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: Fractions, digits, decimal expansion

> Bob said: "9 does not appear".
>
> Do you have a proof, or this just a conjecture? [I seem to be asking that
> a
> lot recently!
> But in mathematics the distinction is vital.]
>
> Maybe I did not understand the question, but suppose after a while we
> reach
> the number 11111145478227154832
> (since the terms must be distinct, the sequence either dies (unlikely
> here)
> or the terms
> get bigger and bigger, so after a zillion steps we will see large
> numbers).
>
> Then the next term could be 9, because 11111145478227154832/9
> = 1234571719803017203.5555555555...
> and all the digits on the left also appear on the right.
>
> My guess is that every number will appear.
>
>
>
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>
> On Mon, Dec 14, 2015 at 10:07 AM, Bob Selcoe <rselcoe at entouchonline.net>
> wrote:
>
>> Hi Eric an Seqfans,
>>
>> 9 does not appear.
>>
>> Cheers,
>> Bob Selcoe
>>
>> --------------------------------------------------
>> From: "Eric Angelini" <Eric.Angelini at kntv.be>
>> Sent: Monday, December 14, 2015 8:59 AM
>> To: "Sequence Fanatics" <seqfan at list.seqfan.eu>
>> Cc: <alexandre.wajnberg at skynet.be>
>> Subject: [seqfan] Fractions, digits, decimal expansion
>>
>>
>> Hello SeqFans,
>>> Here is an echo to the seq computed by Hans Havermann last July:
>>> https://oeis.org/A257664
>>>
>>> a(1)=1; a(n+1) is the smallest positive integer not yet used
>>> such that the set of the different digits present in the
>>> fraction F=a(n)/a(n+1) is visible in the decimal expansion of F:
>>>
>>> S = 1,6,13,10,14,17,7,8,...
>>>
>>> Example:
>>> 1/6 = 0,1666... (we see 1 and 6)
>>> 6/13 = 0,461538461538... (we see 6, 1 and 3)
>>> 13/10 = 1,30 (yes, we admit this zero)
>>> 10/14 = 0,7142857142... (we see 1, 0 and 4)
>>> 14/17 = 0,8235294117... (we see 1, 4 and 7)
>>> 17/7 = 2,4285714285... (we see 1 and 7)
>>> 7/8 = 0,875 (we see 7 and 8)
>>> etc.
>>>
>>> Is S a permutation of the natural numbers > 0?
>>>
>>> Best,
>>> É.
>>>
>>>
>>>
>>> _______________________________________________
>>>
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>>>
>> _______________________________________________
>>
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>>
>
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>
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```

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