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

Author

M. F. Hasler, Feb 01 2016

Keywords

• nonn
• base

Comments

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".

Links

• 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.

Crossrefs

Cf. A005836, A033042.

Programs

Maple

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

Mathematica

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 *)

PARI

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

PARI

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

Formula

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