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

Ignacio Larrosa Cañestro ilarrosa at mundo-r.com
Wed Aug 26 22:46:49 CEST 2009

```Tanya Khovanova wrote:
> I am very sorry.
>
> I confused it with another problem: the number formed by the first k
> integers is divisible by k.
>

Do you meant 381654729(0)?

Saludos,

Ignacio Larrosa Cañestro
A Coruña (España)
ilarrosa at mundo-r.com

> --- On Wed, 8/26/09, Robert Israel <israel at math.ubc.ca> wrote:
>
>> From: Robert Israel <israel at math.ubc.ca>
>> Subject: [seqfan] Re: 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, 2:32 PM
>> 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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