Value a(6) in doubt for a =

[Maximilian Hasler]

In http://www.research.att.com/~njas/sequences/A101877 ,
which could be more concisely be defined by:

%N or
%F A101877 a(n) = min { max S | S c N*, sum( 1/x, x in S) = n }

njas writes:

Little is known about the number of different sets S that achieve
a(n). Paul Hanna asks if it is always true that a solution set S for
n+1 must necessarily contain a solution set for n as a subset. This is
true for small n, apparently, but seems to me unlikely to hold in
general. - njas, Dec 31 2005

ME:
The number of sets S such that sum( 1/x, x in S) = n  is clearly infinite :
one can always write the last fraction (1/k = 1/a(n) initially) as a
sum of two different reciprocals with greater denominator :
1 / k = 1/[ k+1 ] + 1/[ k (k+1) ]

Maximilian

On Jan 14, 2008 12:46 PM, Rainer Rosenthal <r.rosenthal at web.de> wrote:
> The values of
> are defined as  a(n) = max(S_n) where the sum of
> reciprocals of S_n is n. It is claimed that no S_n
> with smaller maximum could be found.
>
> I checked
>
>         ReciprocalSum(S_n) = n
>
> for n=1..6 and it was correct for n=1..5 but incorrect
> for n=6:
>
>     ReciprocalSum(S_6) is less than 6 (about 5.76).
>
> I wasn't able to guess what's wrong with S_6 but following
> a remark about the inclusion of S_n in S_{n+1} I checked
> whether S_5 was a subset of the S_6, given there. An no,
> it is not. S_6 lacks {80, 112, 144, 154, 165, 175, 176},
> but adding these is not enough.
>
> How to repair the description of A101877 and confirm
> a(6) = 469?
>
> Rainer Rosenthal
> r.rosenthal at web.de
>
>
>
>



-- 
Maximilian F. Hasler (Maximilian.Hasler(AT)gmail.com)