A146025 -id:A146025 - cp's OEIS frontend

A268337 Numbers which have only digits 0 and 1 in bases 3 and 5.

0, 1, 30, 31, 756, 3250, 3276, 3280, 81255, 81256, 81280, 81900, 81901, 82000, 59078250, 59078251, 59078280, 59078281, 31789468750, 31789468776, 31789469505, 31789469506, 31789471900, 31789471905, 31789471906, 31789472005, 946095722031, 946095800025, 946095800026, 946095800031, 946095800130

Offset: 1

M. F. Hasler, Feb 01 2016

The number 82000 is famous for having only digits 0 and 1 in all bases <= 5, no other such number > 1 is known. See also A146025 and A258981.
If explicit formulas for (convenient) infinite subsequences of this one can be found, this could open new ways to progress on this problem.
The terms come in groups having roughly the first half (or at least third) of digits in common, see the link "Terms in base 10, 5 and 3".

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 84 terms from M. F. Hasler, next 31 terms from Charles R Greathouse IV)
M. F. Hasler, Terms in base 10, 5 and 3.

Cf. A005836, A033042.

d:= 20: # to get all terms < 5^d
res:= NULL:
T:= combinat:-cartprod([[$0..1]$d]):
while not T[finished] do
  r:= T[nextvalue]();
  v:= add(r[i]*5^(d-i),i=1..d);
  if max(convert(v,base,3)) <= 1 then
    res:= res,v
  fi
od:
res; # Robert Israel, Feb 01 2016

Module[{t=Tuples[{0,1},25],b3,b5},b3=FromDigits[#,3]&/@t;b5=FromDigits[ #,5]&/@t;Intersection[b3,b5]] (* The program generates the first 26 terms of the sequence. *) (* Harvey P. Dale, Dec 13 2021 *)

print1(0);for(n=1,1e10,vecmax(digits(t=subst(Pol(binary(n)),'x,5),3))<2&&print1(","t))

list(lim)=my(v=List([0]),d=digits(lim\1,5),t); for(i=1,#d, if(d[i]>1, for(j=i,#d, d[j]=1); break)); for(n=1,fromdigits(d,5), t=fromdigits(binary(n),5); if(vecmax(digits(t,3))<2, listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Feb 02 2016

a(n) >> n^k with k = log 5/log 2 = 2.321928.... - Charles R Greathouse IV, Feb 02 2016

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

Original entry on oeis.org

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

Offset: 1

Robert Israel, Oct 23 2015

Intersection of A000695 and A033042.

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

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

Cf. A000695, A033042, A146025, A258981.

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));

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

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

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A268337 Numbers which have only digits 0 and 1 in bases 3 and 5.

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

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Maple

Mathematica

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