Almost Integer

[Gerald McGarvey]

While on the subject of almost integers, here's some detail (although not 
an explanation)
regarding log(Pi^4 + Pi^5) = 5.99999995619189332962....

   log(Pi^4 + Pi^5) = 4*log(Pi) + log(1 + Pi)

   The continued fraction for 4*log(Pi)   begins [4;1,1,2,1,2,99,5,...
   The continued fraction for log(1 + Pi) begins   [1;2,2,1,2,99,2,...
   If the convergents that include terms up to 99 are added, the sum is 
1017/712 + 3255/712 = 6.
   The same type of pattern holds for 3*log(Pi) + log(Pi + Pi^2) etc.
   I'm thinking of submitted some entries related to this.

   log(1 + Pi) (OEIS entry A085668) also has this curiosity (c = log(1 + Pi)):

     c^(c^2) = 2.033333996147931...

       with a CF beginning [2;29,1,1675,2,...
       the log10(max(CF)) is 3.2240148..., the keenness depends on if 
log(1+Pi) is counted
       as a singular constant and if each occurrence is counted.

     c^(c^2) * log(Pi^4 + Pi^5) = 12.200003887811074972568...

Something else, somewhat akin to .4^.4:

  2.2^2.2 = 5.6666957787500789138269732...  (log10(max(CF)) = 3.5816...)
  exp(2.2^2.2) = 289.07777765843044070026469...  (log10(max(CF)) = 3.0145...)

Regarding the Feigenbaum delta constant d:

Gamma(d)^2 = 217.99997644999881... (A102817)
(Gamma(d)/d)^2 = 9.999336731332... (A102819) d - log10(d) = 3.99995898...
H(2,d) - 10/d = 1.10000013419... (H(2,d) is the Heronian mean of 2 and d)

Gerald

At 10:55 AM 5/5/2005, Ed Pegg Jr wrote:
>I recently did a column on this topic.
>http://www.maa.org/editorial/mathgames/mathgames_03_15_05.html
>
>I think my favorite item here is the fourth root of 9.1, which
>is subtly related to 
>http://www.research.att.com/projects/OEIS?Anum=A002072 -- I was
>dismissive of 9.1 at first, but NJA liked it. He wound up
>being correct that it was related to another sequence.
>
>Ed Pegg Jr.
>
>Dean Hickerson wrote:
>>Yasutoshi Kohmoto wrote:
>>
>>>Recently Mathworld's description about "Almost Integer" was updated.
>>>I expected my example which I had mailed to Eric Weisstein was described
>>>on it.
>>>         Pi^14/(9103887*Zeta(9))=1.00000000004539
>>>But he didn't do so.
>>>
>>>    Is it not interesting?
>>
>>Not very.  Suppose you compute all numbers of the form
>>     pi^ab / (cdefghi * zeta(j))
>>where a,b,...,j are decimal digits.  There are 8999999100 choices for the
>>digits, so it's not surprising that one of them has 10 zeroes after the
>>decimal point.  If you found one that had a lot more than 10 zeroes, that
>>might be interesting.  (E.g., pi^8/(9450 * zeta(8)) = 1.000000000000..., but
>>here we already know that it's exactly 1.)
>>Another way to write your equation is:  pi^14/zeta(9) = 9103887.0004132...
>>There are 900 choices for the digits in  pi^ab/zeta(c),  so again it's not
>>surprising that there's one with 3 zeroes after the decimal point.
>>Dean Hickerson
>>dean at math.ucdavis.edu