Date: Thu, 24 May 2007 21:56:35 -0400
From: Keith Schneider (schneidk(AT)email.unc.edu)
To: njas@research.att.com
Subject: Re:  A052065-8

The following Mathematica code generates the first 25 terms of each of 
A052065-8. It runs in about ten minutes on my machine. 

Remove[A052065Term, A052067Term]
ConsecDigitsQ[diglist_] := Piecewise[{{(! (diglist[[1]] == diglist[[2]])) && \
ConsecDigitsQ[Delete[diglist, 1]], Length[diglist] > 1}}, True];
OneOffDigitsQ[diglist_] := Piecewise[{{(! (
            diglist[[1]] == diglist[[3]])) && OneOffDigitsQ[Delete[diglist, \
1]], Length[diglist] > 2}}, True];
PalFreeQ[n_] :=      ConsecDigitsQ[IntegerDigits[n]] && 
OneOffDigitsQ[IntegerDigits[n]];
StartVal65[n_] := Floor[Floor[(10^(2n)*340/333)]^(1/2)];
A052065Term[n_] := A052065Term[n] = Module[{
            i}, i = StartVal65[n]; While[! PalFreeQ[i^2], i++]; i];
A052066Term[n_] := A052065Term[n]^2;
StartVal67[n_] := Floor[Floor[(10^(3n)*340/333)]^(1/3)];
A052067Term[
    n_] := A052067Term[n] = Module[{i}, i = StartVal67[n]; While[!      
PalFreeQ[i^3], i++]; i];
A052068Term[n_] := A052067Term[n]^3;
Table[A052065Term[i], {i, 1, 25}] // TableForm
Table[A052066Term[i], {i, 1, 25}] // TableForm
Table[A052067Term[i], {i, 1, 25}] // TableForm
Table[A052068Term[i], {i, 1, 25}] // TableForm