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

Tanya Khovanova mathoflove-seqfan at yahoo.com
Wed Aug 26 20:41:00 CEST 2009

```I am very sorry.

I confused it with another problem: the number formed by the first k integers is divisible by k.

I am sorry.

--- 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/
> >>
> >
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
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