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A258107 Smallest number > 1 whose representation in all bases up to n consists only of 0's and 1's. 11
2, 3, 4, 82000 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,1
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)
LINKS
Thomas Oléron Evans, Solution: Covering all the bases
James Grime and Brady Haran, Why 82,000 is an extraordinary number, Numberphile video, 2015.
EXAMPLE
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
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 *)
CROSSREFS
Cf. A146025.
Sequence in context: A038105 A143716 A228311 * A307256 A107656 A330930
KEYWORD
nonn,base,more
AUTHOR
STATUS
approved

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Last modified May 19 14:45 EDT 2024. Contains 372698 sequences. (Running on oeis4.)