A263684 Numbers whose base-4 and base-5 representations have only 0's and 1's.

0, 1, 5, 16400, 16401, 16405, 82000, 82001, 82005

Offset: 1

Author

Robert Israel, Oct 23 2015

Keywords

• nonn
• fini
• full

Comments

Intersection of A000695 and A033042.

These appear to be all the terms. There are no more below 10^500.

Examples

			16400 is 10000100 in base 4 and 1011100 in base 5.
		

Crossrefs

Cf. A000695, A033042, A146025, A258981.

Programs

Maple

split:= proc(ab, B)
   local a,b,La, Lb, k, j, a1,  a2, b1, b2, x;
   global Res, count;
   a:= ab[1]; b:= ab[2];
   if b-a <= 1000 then
      for x from a to b-1 do
        if max(convert(x,base,4)) <= 1 and max(convert(x,base,5)) <= 1 then
           count:= count+1; Res[count]:= x
        fi
      od;
      return ({});
   fi;
   La:= convert(a,base,B);
   Lb:= convert(b,base,B);
   if nops(Lb) > nops(La) then La:= [op(La),0$(nops(Lb)-nops(La))] fi;
   k:= ListTools:-SelectLast(`>`,Lb-La,0,output=indices);
   if La[k] = 0 then
     a1:= a;
     b1:= 2 + add(B^i,i=1..k-2) + add(La[i]*B^(i-1),i=k+1..nops(La));
     a2:= B^(k-1) + add(La[i]*B^(i-1),i=k+1..nops(La));
     b2:= min(b, b1 + B^(k-1));
     return(select(t -> (t[1] t[1] 0 do
   Cands:= map(op@split, Cands, 5);
   Cands:= map(op@split, Cands, 4);
od:
sort(convert(Res,list));

Mathematica

Select[Range[0,83000],Max[Join[IntegerDigits[#,4],IntegerDigits[#,5]]]<2&] (* Harvey P. Dale, Sep 04 2018 *)

PARI

isok(n) = (n==0) || ((vecmax(digits(n,4))<=1) && (vecmax(digits(n,5))<=1)); \\ Michel Marcus, Oct 24 2015