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%I #19 Apr 01 2024 20:10:41
%S 0,1,2,3,4,5,6,7,8,9,10,20,30,33,40,60,66,80,90,99,252,272,292,313,
%T 330,585,626,660,717,990,2112,2720,2772,2920,4224,5850,6336,7447,7470,
%U 8448,8580,9009,15351,21120,22122,25752,32223,39993,40904,42240,44244,48384
%N Numbers that are palindromic in base 2 as well as in base 10 (initial zeros may be prepended).
%H Robert Israel, <a href="/A069024/b069024.txt">Table of n, a(n) for n = 1..243</a>
%e 66 in base 2 is 1000010, which is palindromic if rewritten as 01000010.
%p nextpal:= proc(p,d,V,b)
%p local i,i2,pp,m,m2;
%p pp:=p;
%p V[1]:= V[1]+1;
%p m2:= floor(d/2);
%p i2:= ceil(d/2);
%p if d::odd then pp:= pp + b^m2 else pp:= pp + b^m2 + b^(m2-1) fi;
%p for i from 1 while V[i] = b do
%p V[i]:= 0:
%p if i = i2 then
%p if d::even then
%p ArrayTools:-Extend(V,[1],inplace);
%p return b^d+1, d+1, V
%p else
%p V[i2]:= 1;
%p return b^d+1, d+1, V;
%p fi;
%p fi;
%p V[i+1]:= V[i+1]+1;
%p if (d::odd and i=1) then pp:= pp + b^(i2-i-1) else
%p pp:= pp + b^(i2-i-1) - b^(i2-i+1) fi;
%p od;
%p return pp, d, V
%p end proc:
%p count:= 1:
%p S:= 0:
%p p2[0]:=1: V2[0]:= <1>: d2[0]:= 1:m2:= 0:
%p p10[0]:= 1: V10[0]:= <1>: d10[0]:= 1: m10:= 0:
%p while count < 100 do
%p i2:= min[index]([seq(p2[i],i=0..m2)])-1; p2o:= p2[i2];
%p i10:= min[index]([seq(p10[i],i=0..m10)])-1; p10o:= p10[i10];
%p if p2o = p10o then
%p S:= S, p2o; count:= count+1;
%p fi;
%p if p2o <= p10o then x, d2[i2], V2[i2]:= nextpal(p2o/2^i2, d2[i2], V2[i2],2); p2[i2]:= 2^i2 *x;
%p if i2 = m2 then m2:= m2+1; p2[m2]:= 2^m2; V2[m2]:= <1>; d2[m2]:= 1;
%p fi;
%p else
%p x, d10[i10], V10[i10]:= nextpal(p10o/10^i10, d10[i10], V10[i10],10);
%p p10[i10]:= 10^i10 * x;
%p if i10 = m10 then m10:= m10+1; p10[m10]:= 10^m10; V10[m10]:= <1>; d10[m10]:= 1
%p fi fi od:
%p S; # _Robert Israel_, Apr 01 2024
%t pal[n_, b_] := (z=IntegerDigits[n, b]) == Reverse[z]; extpal[n_, b_] := If[Mod[n, b]>0, pal[n, b], extpal[n/b, b]]; Select[Range[50000], extpal[ #, 10]&&extpal[ #, 2]&]
%Y Cf. A007632.
%Y Intersection of A061917 and A057890.
%K nonn,base
%O 1,3
%A _Amarnath Murthy_, Apr 02 2002
%E Edited by _Dean Hickerson_, Apr 06 2002
%E 0 inserted by _Sean A. Irvine_, Mar 29 2024
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