A258107 Smallest number > 1 whose representation in all bases up to n consists only of 0's and 1's.

2, 3, 4, 82000

Offset: 2

Author

Bernardo Boncompagni, May 20 2015

Keywords

• nonn
• base
• more

Comments

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

From Devansh Singh, Mar 14 2021: (Start)

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.

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.

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)

Examples

			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.
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
		

Links

• Thomas Oléron Evans, Solution: Covering all the bases
• Richard Green, A Curious Property of 82000
• James Grime and Brady Haran, Why 82,000 is an extraordinary number, Numberphile video, 2015.

Crossrefs

Cf. A146025.

Programs

Mathematica

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 *)