Mismatch between A097398 and UPiNT 3rd ed.

[Richard Guy]

I've had a message from Henry, and any errors
are probably due either to my inaccurate
transcription, or possibly to typos by
Fibonacci Quarterly.  As the years go by
computers, both electronic and human,get
better and better.  Perhaps some keen type
will check the table below, and even extend
it?   Many thanks in anticipation.   R.

On Thu, 2 Dec 2004, Richard Guy wrote:

> I'm copying this message to Sloane's Dream Team
> in the hope that some computer can settle the
> doubtful values (marked ***) in the following
> table:
>
>
> \begin{center}
> \begin{tabular}{lcccccccccc}
> \quad $k$ & 2 &   3 &   4 &   5 &   6 &   7 &  8 & 
> 9 & 10 & 11 \\
> $x_1=2$  & 43 &  89 &  97 & 214 &  19 & 239 & 37 & 
> 79 & 83 & 239 \\
> $x_1=3$  &  7 &  89 &  17 &  43 &  83 & 191 &  7 & 
> 127 & 31 & 389 \\ *** 338?
> $x_1=4$  & 17 &  89 &  23 & 139 &  13 & 359 & 23 & 
> 158 & 41 & 239 \\ *** 139?
> $x_1=5$  & 34 &  89 &  97 & 107 &  19 & 419 & 37 & 
> 79 & 83 & 137 \\
> $x_1=6$  & 17 &  31 & 149 & 269 &  13 & 127 & 23 & 
> 103 & 71 & 239 \\
> $x_1=7$  & 17 & 151 &  13 & 107 &  37 & 127 & 37 & 
> 103 & 83 & 239 \\
> $x_1=8$  & 51 &  79 &  13 & 214 &  13 & 239 & 17 & 
> 163 & 71 & 239 \\
> $x_1=9$  & 17 &  89 &  83 & 139 &  37 & 191 & 23 & 
> 103 & 23 & 169 \\ *** 239?
> $x_1=10$ &  7 &  79 &  23 & 251 & 347 & 239 &  7 & 
> 163 & 41 & 239 \\
> $x_1=11$ & 34 & 601 &  13 & 107 &  19 & 478 & 37 & 
> 79 & 31 & 389
> \end{tabular}                           ***
> \end{center}                            461?
>
> Many thanks in anticipation of help.     R.
>
> On Fri, 17 Sep 2004, Hugo Pfoertner wrote:
>
>> Dear Neil, (CC Richard Guy, Henry Ibstedt),
>> 
>> one month ago I had submitted A097398
>> http://www.research.att.com/projects/OEIS?Anum=A097398
>> 
>> I just copied it from the URL you had posted 
>> earlier, asking if someone
>> could find interesting stuff in Henry Ibstedt's 
>> book "Mainly Natural
>> Numbers" [1]. At the time when I prepared A097398, 
>> the on-line version
>> of [1] had disappeared, so I didn't include a link 
>> to it in the
>> sequence.
>> Currently it is available again, but I don't know 
>> if that will last
>> longer ...
>> http://www.gallup.unm.edu/~smarandache/Ibstedt-Book3.pdf
>> 
>> I now compared the table given on page 35 of [1] 
>> that is reproduced in
>> the comment section of A097398 against the 
>> transposed version given in
>> Richard's "Unsolved Problems..., 3rd edition" [2], 
>> page 329, E15 A
>> recursion of Goebel. Whereas the parts k<11 and 
>> x_1<11 are identical
>> (using Richard's notation, the last row (x_11=11) 
>> and the last column
>> k=11 show differences:
>> 
>> x_1=11 [1]: 34 601  13 107  19 461  37  79  31 389 
>> Ibstedt
>>       [2]: 34 601  13 107  19 478  37  79  31 389 
>> rkg
>>                               ^^^
>> k=11   [1]: 239 338 139 137 239 239 239 239 239 
>> 389 Ibstedt
>>       [2]: 239 389 239 137 239 239 239 169 239 389 
>> rkg
>>                ^^^ ^^^                 ^^^
>> 
>> I don't know how reliable Henry Ibstedt's results 
>> are. I did no own
>> computations, except for the 
>> 2,3,5,10,28,154,3520,.. Goebel's sequence.
>> The fact that Ibstedt gives a wrong version of 
>> this sequence in [1],
>> 
>> 2,3,5,10,28,154,3520,_15518880_,267593772160,_160642690122633501504
>> instead of
>> 2,3,5,10,28,154,3520, 1551880,  267593772160, 
>> 7160642690122633501504
>> (A003504)
>> 
>> casts some doubts on the reliability of the 
>> reproduction of numerical
>> results in [1]. Looks like being copied manually. 
>> I have no idea how to
>> resolve the mismatch mentioned above. Fortunately 
>> the part of the
>> rectangular array given in the %S%T%U part of the 
>> sequence is not
>> affected. I do not want to dive deeper into this 
>> subject, but maybe
>> Henry Ibstedt himselve (I CCed this message to 
>> him) can resolve the
>> question. Of course it would be best to 
>> recalculate his results
>> independently.
>> 
>> (BTW: the initialization of A003504 and the 
>> related sequences is at
>> least problematic, as it is described currently - 
>> division by zero
>> results. I'll make an recommendation in a comment 
>> on A003054 ).
>> 
>> Best wishes to all
>> 
>> Hugo Pfoertner