Q: definition of A134592

[Richard Mathar]

Based on David Cantrell's heuristic formula for A136617      
I computed the first 10000 entries of A081881.
While A136617 tells us how many items (egyptian fractions
starting with 1/n) can be packed into a bin of length 1,
a = A081881 helps putting all egyptian fractions into bins.
The n-th bin is filled with 1/a(n) up to, but not including,
1/a(n+1). Each bin is filled as far as possible,
i.e. 1/a(n+1) cannot be put into the n-th bin and it is the
first egyptian fraction having this property.

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Gerald McGarvey observed that

         d(n) = floor(e*a(n)) - a(n+1)             (1)

has the only values d=0 or d=1.

There is a certain interesting relationship between (1) and
the "fullness F(n)" of the n-th bin, as packed according to
the description of a = A081881.
We add all egyptian fractions starting with 1/a(n) up to, but
not including a(n+1) to fill the n-th bin. This sum doesn't
exceed 1, but adding 1/a(n+1) does. So we can write

     a(n+1)-1
     --------
     \         1      1       1
      >       ---  + ---- * ------   =   1         (2)
     /         i     F(n)   a(n+1)
     --------
     i = a(n)

where F(n) > 1 is related to the "fullness" of the n-th bin.
Large F correspond to heavily filled bins. Not only doesn't
1/a(n+1) fit into the n-th bin, but there is room for only a
tiny part of it.

Using the first 10000 elements of A081881 I found out that
F0 ~ 7.1 is a critical factor with respect to Gerald McGarvey's
observation (1). There seems to be some critical value F0
bounded by

       7.098156  <   F0  <   7.106632              (3)

such that d(n) = 0 is equivalent to F(n) > F0. The values n
corresponding to (3) are n=2748 and n=5925.

These extreme values together with the other extremes encoun-
tered during my investigation of the first 10000 values of
A081881 are listed in the following table:

      n       1210      2748       5925        9953
    ----------------------------------------------------
     d(n)       1         1          0           0
     F(n)   1.000009  7.098156   7.106632    7952.60277

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It would be interesting whether my observations could shed
some light on the validity of David Cantrell's interesting
heuristic formula for A136617.
Another interesting thing is the meaning and exact value of F0.

Best regards,
Rainer Rosenthal
r.rosenthal at web.de