A258946 Numbers that can be expressed using only the digits 0 and 1 in no more than three different bases.

2, 3, 4, 5, 6, 7, 8, 11, 14, 15, 18, 19, 22, 23, 24, 29, 32, 33, 34, 35, 38, 41, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 58, 59, 60, 61, 62, 63, 66, 67, 70, 71, 74, 75, 76, 77, 78, 79, 83, 86, 87, 88, 89, 92, 95, 96, 97, 98, 99, 102, 103, 104, 105, 106, 107

Offset

1,1

Comments

All integers n >= 4 may trivially be expressed using only the digits 0 and 1 in three different bases: 2, n-1 (as '11') and n (as '10'). The numbers in this sequence cannot be expressed using only 0 and 1 in any other base.

The only positive integers that may be expressed using only the digits 0 and 1 in fewer than three different bases are 2 and 3, for which the values {2, n-1, n} are not all distinct or are not all valid bases.

An equivalent definition: For each term a(n) of this sequence, there are at most three integers k >= 2 for which a(n) is a sum of distinct nonnegative integer powers of k.

Links

Thomas Oléron Evans, Table of n, a(n) for n = 1..10000

Thomas Oléron Evans, Python program

Example

5 is a term of the sequence, because 5 may be expressed using only the digits 0 and 1 in precisely three different bases: 2, 4 and 5 (5 is '12' in base 3).
9 is not a term of the sequence, because 9 can be expressed using only the digits 0 and 1 in four different bases: 2, 3, 8, 9 (9 is '100' in base 3).

Maple

filter:= proc(n)
  local b;
  for b from 3 to n-2 do
    if max(convert(n, base, b)) <= 1 then return false
    fi
  od:
true
end proc:
select(filter, [$2..1000]); # Robert Israel, Jun 19 2015

Prog

(PARI) is(n)=if(n<2, return(0)); for(b=3, sqrtint(n), if(vecmax(digits(n, b))<2, return(0))); 1 \\ Charles R Greathouse IV, Jun 15 2015

Crossrefs

Subsequence of A074940.

Cf. A146025, A258107, A005836, A000695, A033042.

Sequence in context: A067133 A192588 A351914 * A008815 A298434 A200371

Adjacent sequences: A258943 A258944 A258945 * A258947 A258948 A258949

Keyword

nonn,base

Author

Thomas Oléron Evans, Jun 15 2015

Status

approved