A258946 - OEIS

%I #21 Jul 19 2015 07:26:00

%S 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,

%T 47,48,51,52,53,54,55,58,59,60,61,62,63,66,67,70,71,74,75,76,77,78,79,

%U 83,86,87,88,89,92,95,96,97,98,99,102,103,104,105,106,107

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

%C 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.

%C 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.

%C 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.

%H Thomas Oléron Evans, <a href="/A258946/b258946.txt">Table of n, a(n) for n = 1..10000</a>

%H Thomas Oléron Evans, <a href="/A258946/a258946.txt">Python program</a>

%e 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).

%e 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).

%p filter:= proc(n)

%p   local b;

%p   for b from 3 to n-2 do

%p     if max(convert(n,base,b)) <= 1 then return false

%p     fi

%p   od:

%p true

%p end proc:

%p select(filter, [$2..1000]); # _Robert Israel_, Jun 19 2015

%o (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

%Y Subsequence of A074940.

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

%K nonn,base

%O 1,1

%A _Thomas Oléron Evans_, Jun 15 2015