[seqfan] EPRNs

[Hans Havermann]

Last week, I started looking at the ten or so < https://oeis.org/search?q=eprn 
  > sequences based on S.S. Gupta's EPRNs (see < https://oeis.org/A066531 
  > for the basic construct) because they were related to the number- 
times-reversal < https://oeis.org/A061205 > sequence that Franklin T.  
Adams-Watters mentioned here on January 7, a sequence with which I am  
still somewhat involved.

I noticed a number of things wrong, here and there, and I have been  
submitting "corrections" and added a few b-files where I saw fit.  
Yesterday I noticed that some of my corrections required their own  
corrections and today I resolved to put my empirical meanderings on a  
firmer foundation. I was a little shocked by what I found...

Earlier today I plugged 10^8 into the Mathematica formula for A066531,  
netting me over five million terms. I felt safe submitting the first  
10000 of these terms which showed that the 10000-term b-file I  
submitted a couple of days ago (based on a smaller plug-in) was  
incorrect. Well, I now believe that only 8000 terms may actually be  
correct. This is deduced from plugging into the formula 10^3, 10^4,  
10^5, 10^6, 10^7 and seeing just how many of the terms actually agree  
with the next-higher-power calculation:

10^3: only 18 of the 40 terms can be used (up to 24300)
10^4: only 60 of the 468 terms can be used (up to 243000)
10^5: only 276 of the 5068 terms can be used (up to 2473240)
10^6: only 736 of the 50630 terms can be used (up to 25018420)
10^7: only 3132 of the 505022 terms can be used (up to 251652240)

This extrapolates to about 8075 terms of the 5027910 terms for 10^8  
that can be used. I have insufficient memory to calculate 10^9 and an  
insufficient understanding of the mathematics that appears to limit  
these calculations so dramatically. What's worse is that some of the  
other EPRN sequences derived from A066531 use terms way, way beyond  
these 8000 and that, therefore, there is a good probability of some  
failure in their current form. I'm a little sorry now that I made  
submissions without fully understanding these sequences' limitations.

I'm throwing in the towel. Perhaps a new pair of eyes (and a  
discussion here) can help these sequences along.