[seqfan] Re: Pieces of cake sequence A265286 and a question

[Neil Sloane]

Max, let me reveal what I really wanted!

The original problem is:

Given n, look for the smallest set of fractions {f_1, f_2, ..., f_M} in the
range 0 to 1 such that for each k with 1 <= k <= n, we can partition the
f_i into k groups whose sums are equal. For n=5 the minimal M is 9, and a
solution is
{1/60, 1/30, 1/20, 1/12, 7/60, 2/15, 1/6, 1/5, 1/5}

This involves fractions. If we multiply through by the LCM of the
denominators we get an equivalent
problem in integers: find the smallest number of integers F_1, ..., F_M
(meaning minimize M)
such that they can be partitioned into k clumps with equal sums for all k =
1..n.

Look at all solutions for a given n, and define g(n) to be the min value of
max{F_1, ...,f_M}.

Then g(1)=1, g(2)=2 [use {1,1}], g(3)=2 [use {1,1,2,2}].  Can you supply
any other values of g(n)?

>From your example in A265286 for n=5 with M=9, we have g(5) <= 60.

Your example for n=6, M=11,
1/30, 1/30, 1/20, 1/15, 1/15, 1/12, 1/12, 7/60, 2/15, 1/6, 1/6]
shows that g(6) <= 60.



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 Sun, Jan 24, 2016 at 2:54 PM, Max Alekseyev <maxale at gmail.com> wrote:

> Ron, gurobi uses all available cores (40 in my case) by default; I had
> nothing to do here.
>
> On Sun, Jan 24, 2016 at 2:42 PM, Ron Hardin <rhhardin at att.net> wrote:
>
> > My experience is that running two cores on a single machine slows it down
> > a lot, owing to memory cache conflicts.
> > It probably depends on how you access memory.  If it's mostly local per
> > process, it may not hurt much. rhhardin at mindspring.com rhhardin at att.net
> > (either)
> >
> >
> >
> >       From: Max Alekseyev <maxale at gmail.com>
> >  To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> >  Sent: Sunday, January 24, 2016 10:33 AM
> >  Subject: [seqfan] Re: Pieces of cake sequence A265286 and a question
> >
> > For n=6, the minimum smallest piece in an 11-piece solution equals 1/120,
> > like in
> > [1/120, 1/40, 1/30, 7/120, 3/40, 11/120, 13/120, 1/8, 17/120, 1/6, 1/6]
> >
> > It would take about 37 days of single-core computations in Sage+Gurobi.
> > Luckily I run on a 40-core system and it took only about a day to
> establish
> > the above result.
> >
> > Regards,
> > Max
> >
> >
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>