Value a(6) in doubt for a = A101877

[Rainer Rosenthal]

hv at crypt.org wrote:
> 
> Apologies for the inconvenience, and thanks for spotting the error.
> 

You are welcome. I checked the corrected set and it's OK. The problem
is very nice and closely related to what I became interested in recently.
I looked for given n for maximal m such that the reciprocals of the
set { n, n+1, ... , m} have a sum not exceeding 1. This has become
sequence A136616 and is already submitted together with the related
sequence A136617. In our German newsgroup de.sci.mathematik I told
about these problems and put a little ASCII figure for visualizing
the fun with A136616 and A136617:

:
:                                                         _
:                                                     ___( )___
:         1   1        1                             |         |
:         -  ---       -                             |    1    |
:      \  n  n+1  ...  m      /               \      |         |     /
:       \____________________/                 \_____|_________|____/
:                 |___________________o___________________|
:                               _____/ \_____
:                         _____(             )_____
:                        (                     RwR)
:   ---------------------------------------------------------------------
:
:________________________________________________________________________
:
: Figure 1:   The "harmonic balance"
:             Find largest m with left side up.
:

In the case of A101877 there is "n" on one side and has to be balanced by
as few egyptian fractions as possible on the other side: 

:
:                 _ 
:             ___( )___
:            |         |                          1   1        1
:            |    n    |                          -  ---      ----
:     \      |         |     /                \   x   y  ...   z     /
:      \_____|_________|____/                  \____________________/
:                 |___________________o___________________|
:                               _____/ \_____
:                         _____(             )_____
:                        (                     RwR)
:   ---------------------------------------------------------------------
:
:________________________________________________________________________
:
: Figure 2:   The "egyptian balance"
:             Find smallest set of egyptian fractions for equilibrium.
:

It seems reasonable that my description agrees with your definition, but
in fact it is a conjecture:

     RR-conjecture for A101877
     ===========================================================
     Let S_n be a smallest set such that ReciprocalSum(S_n) = n.
     Then max(S_n) = A101877(n).

BTW the sizes card(S_n) are:  1,4,13,37,110,291,...

They are not in the OEIS yet and it would be nice if you submitted them
as a new sequence. Judging from the other OEIS sequences starting with
1,4,13,37 I would expect something like 750 for your s(7) if you will
ever find it (good luck to you!)

Cheers,
Rainer




           




Rainer Rosenthal <r.rosenthal at web.de> wrote:
[...]
:In the case of A101877 there is "n" on one side and has to be balanced by
:as few egyptian fractions as possible on the other side: 
:
::
::                 _ 
::             ___( )___
::            |         |                          1   1        1
::            |    n    |                          -  ---      ----
::     \      |         |     /                \   x   y  ...   z     /
::      \_____|_________|____/                  \____________________/
::                 |___________________o___________________|
::                               _____/ \_____
::                         _____(             )_____
::                        (                     RwR)
::   ---------------------------------------------------------------------
::
::________________________________________________________________________
::
:: Figure 2:   The "egyptian balance"
::             Find smallest set of egyptian fractions for equilibrium.
::
:
:It seems reasonable that my description agrees with your definition, but
:in fact it is a conjecture:
:
:     RR-conjecture for A101877
:     ===========================================================
:     Let S_n be a smallest set such that ReciprocalSum(S_n) = n.
:     Then max(S_n) = A101877(n).
:
:BTW the sizes card(S_n) are:  1,4,13,37,110,291,...
:
:They are not in the OEIS yet and it would be nice if you submitted them
:as a new sequence. Judging from the other OEIS sequences starting with
:1,4,13,37 I would expect something like 750 for your s(7) if you will
:ever find it (good luck to you!)

I don't think your conjecture here is particularly likely to be generally
that a set of smaller cardinality but greater maximum exists. Moreover,
my code does not attempt to find the smallest set even for a given maximum,
cardinalities exist with the same maximum, and that the solution found by
my code is not the smallest of those.

For what it's worth, the notes from my previous attempt tell me I found
1200 <= a(7) <= 1243 (ie that I found a set S_7 with max(S_7)=1243, and
proved no such set existed with max(S) < 1200). I have |S_7|=806 for this
}.

Looking at the first composite missing from this set (292, by inspection)
prompts me to look at the multiples of 73, and I quickly find that we
include the subset { 730, 876, 1095 } which could be replaced by { 292 }

Hugo