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

1,3

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.

Formula

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

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 *)

Prog

(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

Crossrefs

Cf. A005836, A033042.

Sequence in context: A042850 A042852 A042854 * A042856 A042857 A042858

Adjacent sequences: A268334 A268335 A268336 * A268338 A268339 A268340

Keyword

nonn,base

Author

M. F. Hasler, Feb 01 2016

Status

approved