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%I #84 Dec 03 2023 15:26:33
%S 2,3,4,82000
%N Smallest number > 1 whose representation in all bases up to n consists only of 0's and 1's.
%C As with A146025, it is a plausible conjecture that there are no more terms, but this has not been proved. - _Daniel Mondot_, Dec 16 2016
%C From _Devansh Singh_, Mar 14 2021: (Start)
%C If a(n) exists then b-1|(N-Sum_{i>=0} A(i)), b-2|(N-Sum_{i>=0} A(i)*2^i), b-3|(N-Sum_{i>=0} A(i)*3^i), ... where b <= n.
%C If a(n) exists for n > 5 then let it be N. N = Sum_{i>=0} A(i)*b^i where A(i) is the i-th digit (0 or 1 only) of N starting from the right in base b <= n.
%C N = Sum_{i>=0} A(i)*b'^i + Sum_{i>=1} A(i)*(b^i - b'^i), where b' < b. If b=6 then we can see that 5|(N-Sum_{i>=0} A(i)), 4|(N-Sum_{i>=0} A(i)*2^i), 3|(N-Sum_{i>=0} A(i)*3^i). (End)
%H Thomas Oléron Evans, <a href="http://www.mathistopheles.co.uk/maths/covering-all-the-bases/solution-covering-all-the-bases/">Solution: Covering all the bases</a>
%H Richard Green, <a href="https://plus.google.com/101584889282878921052/posts/Fni6x2TTeaS">A Curious Property of 82000</a>
%H James Grime and Brady Haran, <a href="https://www.youtube.com/watch?v=LNS1fabDkeA">Why 82,000 is an extraordinary number</a>, Numberphile video, 2015.
%e a(4) = 4 because it is 100 in base 2, 11 in base 3 and 10 in base 4. No smaller number, except 1, can be expressed in such bases with only 0's and 1's.
%e a(5) = 82000: 82000 in bases 2 through 5 is 10100000001010000, 11011111001, 110001100, 10111000, containing only 0's and 1's, while all smaller numbers have a larger digit in one of those bases. For example, 12345 is 11000000111001, 121221020, 3000321, 343340. - _N. J. A. Sloane_, Feb 01 2016
%t Table[k = 2; While[Total[Total@ Drop[RotateRight[DigitCount[k, #]], 2] & /@ Range[3, n]] > 0, k++]; k, {n, 2, 5}] (* _Michael De Vlieger_, Aug 29 2015 *)
%Y Cf. A146025.
%K nonn,base,more
%O 2,1
%A _Bernardo Boncompagni_, May 20 2015
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