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