# [seqfan] Re: 10 integers dividing the sum of the 10 integers

Neil Sloane njasloane at gmail.com
Wed May 15 17:45:51 CEST 2019

We had better say "positive", otherwise {-1,1} is a solution for n=2

Certainly add this to the OEIS, with offset 3,
S = 6, 12, 24, 60, 60, 60, 180, 420, 420, 840, 840,...
or maybe, better,
S = 1, -1, 6, 12, 24, 60, 60, 60, 180, 420, 420, 840, 840,...
with offset 1, adding "or -1 if there is no solution in positive integers.

You should add the triangle too: Irregular triangle read by rows: row n
gives [lex .......], or -1 if there is no solution:

1
-1
1 2 3
1 2 3 6
1 2 3 6 12
1 2 3 4 20 30
1 2 3 4 5 15 30   ...

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.
Email: njasloane at gmail.com

On Mon, May 13, 2019 at 6:14 AM Éric Angelini <bk263401 at skynet.be> wrote:

> Hello SeqFans,
> this was suggested to me by an "Enigma" posted here :
> https://bit.ly/2Vyu9Mv
>
> "Find 10 different integers {a, b, c, ... i, j} such
> that they all divide the sum (a+b+c+d+e+f+g+h+i+j)"
>
> The given solution was not mine -- as I wanted to
> find the lexicographically earliest set of this kind.
>
> Then I started to search for such lexico-sets of
> size "n" (with n > 2). Carole Dubois and I found:
>
> n=3, integers = 1 2 3
> n=4, integers = 1 2 3 6
> n=5, integers = 1 2 3 6 12
> n=6, integers = 1 2 3 4 20 30
> n=7, integers = 1 2 3 4 5 15 30
> n=8, integers = 1 2 3 4 5 10 15 20
> n=9, integers = 1 2 3 4 5 6 9 60 90
> n=10, integers = 1 2 3 4 5 6 7 42 140 210
> n=11, integers = 1 2 3 4 5 6 7 12 30 140 210
> n=12, integers = 1 2 3 4 5 6 7 8 20 84 280 420
> n=13, integers = 1 2 3 4 5 6 7 8 10 24 70 280 420
>
> We stopped there.
>
> Questions:
> Could someone (if this is of interest and not old hat)
> extend this array to, say, n = 100?
> And how could this array enter the OEIS?
> Carine said to me that the successive row-sums might be
> a possibility. We would then have this 11-term start:
>
> S = 6, 12, 24, 60, 60, 60, 180, 420, 420, 840, 840,...
> (this succession is not in the OEIS)
>
> Best,
> É.
>
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>