# [seqfan] Re: Dividing the sum of the k leftmost digits of N by k

Robert Israel israel at math.ubc.ca
Wed Aug 26 20:32:32 CEST 2009

```It can't exist, since for k=10, 0+1+...+9 = 45 is not divisible by 10.
Moreover, if my Maple program is to be believed, there are none with 8 or
9 digits.  However, there are 15 with 7 digits:
1502794
1506394
1560394
2015794
4839605
4893605
4897205
5102794
5106394
5160394
7984205
8439605
8493605
8497205
9784205

Cheers,
Robert Israel

On Wed, 26 Aug 2009, Tanya Khovanova wrote:

> A ten digit number like that exists, and it is unique. I forgot what it is, but I can find it if I have time:
> http://blog.tanyakhovanova.com/?p=31
>
> I believe Martin Gardner wrote about it.
>
> --- On Wed, 8/26/09, Eric Angelini <Eric.Angelini at kntv.be> wrote:
>
>> From: Eric Angelini <Eric.Angelini at kntv.be>
>> Subject: [seqfan]  Dividing the sum of the k leftmost digits of N by k
>> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
>> Date: Wednesday, August 26, 2009, 1:04 PM
>>
>> Hello SeqFans, [idea coming from the recent 'average' post
>> by Zakir]
>>
>> is 978015 the biggest number N with no two same digits
>> having the pro-
>> perty that when the sum of the k leftmost digits of N is
>> divided by k
>> the result is always an integer?
>>
>>
>>
>>    N = 978015
>>
>> - dividing the sum of the 2 leftmost digits by 2: (9+7)/2 =
>> 8
>> - dividing the sum of the 3 leftmost digits by 3: (9+7+8)/3
>> = 8
>> - dividing the sum of the 4 leftmost digits by 4:
>> (9+7+8+0)/4 = 6
>> - dividing the sum of the 5 leftmost digits by 5:
>> (9+7+8+0+1)/5 = 5
>> - dividing the sum of the 6 leftmost digits by 6:
>> (9+7+8+0+1+5)/6 = 5
>>
>> Many seq based on this idea could be added to the OEIS (if
>> of interest)
>>
>> The same with the rightmost digits.
>>
>> (see http://www.research.att.com/~njas/sequences/A061383
>>      "Arithmetic mean of digits is an
>> integer.")
>> Best,
>> É.
>>
>>
>>
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
>
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>
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>
```