Voozh

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

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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

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