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

[Neil Sloane]

Let me try that message again, correcting some errors: This replaces my
previous message.

Max, let me reveal what I really wanted!

The original problem is:

Given n, look for the smallest multiset 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.

Now look at all solutions for a given n (having the minimal number of
fractions), and define g(n) to be the min value of max{F_1, ...,f_M}.

Then g(1)=1, g(2)=1 [use {1,1}], g(3)=2 [use {1,1,2,2}].

>From your example in A265286 for n=5 with M=9, that is,
1/30, 1/30, 1/20, 1/15, 1/15, 1/12, 1/12, 7/60, 2/15, 1/6, 1/6],
after multiplying by the LCM, 60, we get
2,2,3,4,4,5,5,7,8,10,10,
so g(5) <= 10.

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) <= 10.

Can you supply any other values of g(n)?

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