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%I #74 Mar 17 2021 02:54:41
%S 0,1,82000
%N Numbers that can be written in bases 2, 3, 4, and 5 using only the digits 0 and 1.
%C Originally checked to 2^65520 (or about 3*10^19723) on Nov 07 2008. - _Daniel Mondot_, Jan 17 2016
%C Conjectured to be complete. a(4), if it exists, is greater than 10^15. - _Charles R Greathouse IV_, Apr 06 2012
%C Checked to 3125 (5^5) base-5 digits in just under 1/2 hour using a minor modification of the PARI program at A230360. Interestingly, with 5 replaced by 9 and the digits 2 and 3 permitted, it appears the complete set is--somewhat coincidental with this--{0, 1, 2, 3, 8281, 8282, 8283}, see A146026. - _James G. Merickel_, Dec 01 2013
%C Checked to 11 million decimal digits in 1 week using an algorithm that, upon finding that the current guess has non-{0,1} digits in a particular base, increases the guess to only have {0,1} digits in that base. C code in links. - _Alex P. Klinkhamer_, Aug 29 2015
%C It is a plausible conjecture that there are no more terms, but this has not been proved. - _N. J. A. Sloane_, Feb 06 2016
%H Stuart A. Burrell and Han Yu, <a href="https://arxiv.org/abs/1905.00832">Digit expansions of numbers in different bases</a>, arXiv:1905.00832 [math.NT], 2019.
%H Daniel Glasscock, Joel Moreira, and Florian K. Richter, <a href="https://arxiv.org/abs/2007.05480">Additive transversality of fractal sets in the reals and the integers</a>, arXiv:2007.05480 [math.NT], 2020. See p. 5.
%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)
%H Alex P. Klinkhamer, <a href="https://grencez.dev/2015/82000-digits-20150829">Digits of 82000</a>, search algorithm with code and analysis.
%e 82000 = 10100000001010000 (2) = 11011111001 (3) = 110001100 (4) = 10111000 (5).
%t f[n_] := Total[Total@ Drop[RotateRight[DigitCount[n, #]], 2] & /@ Range[3, 5]]; Select[Range[0, 100000], f@ # == 0 &] (* _Michael De Vlieger_, Aug 29 2015 *)
%o (PARI) is(n)=vecmax(digits(n,5))<2 && vecmax(digits(n,4))<2 && vecmax(digits(n,3))<2 \\ _Charles R Greathouse IV_, Aug 31 2015
%Y Intersection of A005836, A000695, and A033042.
%Y Cf. A258981 (bases 2,3,4), A258107 (bases 2..n).
%Y Cf. A131646, A146026, A146027, A230360, A275600.
%K bref,nonn,base
%O 1,3
%A _Daniel Mondot_, Oct 26 2008
%E Edited by _Charles R Greathouse IV_, Nov 01 2009
%E Search limit extended to astronomical odds by _James G. Merickel_, Dec 03 2013
%E Search limit increased again with example code by _Alex P. Klinkhamer_, Aug 29 2015
%E Removed keywords "fini" and "full", since it is only a conjecture that there are no further terms. - _N. J. A. Sloane_, Feb 06 2016
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