A146025 Numbers that can be written in bases 2, 3, 4, and 5 using only the digits 0 and 1.

0, 1, 82000

Offset

1,3

Comments

Originally checked to 2^65520 (or about 3*10^19723) on Nov 07 2008. - Daniel Mondot, Jan 17 2016

Conjectured to be complete. a(4), if it exists, is greater than 10^15. - Charles R Greathouse IV, Apr 06 2012

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

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

It is a plausible conjecture that there are no more terms, but this has not been proved. - N. J. A. Sloane, Feb 06 2016

Links

Table of n, a(n) for n=1..3.

Stuart A. Burrell and Han Yu, Digit expansions of numbers in different bases, arXiv:1905.00832 [math.NT], 2019.

Daniel Glasscock, Joel Moreira, and Florian K. Richter, Additive transversality of fractal sets in the reals and the integers, arXiv:2007.05480 [math.NT], 2020. See p. 5.

James Grime and Brady Haran, Why 82,000 is an extraordinary number, Numberphile video (2015)

Alex P. Klinkhamer, Digits of 82000, search algorithm with code and analysis.

Example

82000 = 10100000001010000 (2) = 11011111001 (3) = 110001100 (4) = 10111000 (5).

Mathematica

f[n_] := Total[Total@ Drop[RotateRight[DigitCount[n, #]], 2] & /@ Range[3, 5]]; Select[Range[0, 100000], f@ # == 0 &] (* Michael De Vlieger, Aug 29 2015 *)

Prog

(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

Crossrefs

Intersection of A005836, A000695, and A033042.

Cf. A258981 (bases 2,3,4), A258107 (bases 2..n).

Cf. A131646, A146026, A146027, A230360, A275600.

Sequence in context: A134122 A138044 A098187 * A361934 A283418 A330872

Adjacent sequences: A146022 A146023 A146024 * A146026 A146027 A146028

Keyword

bref,nonn,base

Author

Daniel Mondot, Oct 26 2008

Extensions

Edited by Charles R Greathouse IV, Nov 01 2009

Search limit extended to astronomical odds by James G. Merickel, Dec 03 2013

Search limit increased again with example code by Alex P. Klinkhamer, Aug 29 2015

Removed keywords "fini" and "full", since it is only a conjecture that there are no further terms. - N. J. A. Sloane, Feb 06 2016

Status

approved