A146027 -id:A146027 - cp's OEIS frontend

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

Daniel Mondot, Oct 26 2008

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

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

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.

Intersection of A005836, A000695, and A033042.
Cf. A258981 (bases 2,3,4), A258107 (bases 2..n).
Cf. A131646, A146026, A146027, A230360, A275600.

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

is(n)=vecmax(digits(n,5))<2 && vecmax(digits(n,4))<2 && vecmax(digits(n,3))<2 \\ Charles R Greathouse IV, Aug 31 2015

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

A146026 Numbers that can be written from base 2 to base 9 using only the digits 0 to 3.

Original entry on oeis.org

0, 1, 2, 3, 8281, 8282, 8283

Offset: 1

Daniel Mondot, Oct 27 2008

Conjectured to be complete.
Checked up to 2^16384 (≈ 1.2*10^4932) on Oct 31 2008. - Daniel Mondot, Jan 17 2016
Checked up to 2^65520 (≈ 3*10^19723) on Nov 13 2008. - Daniel Mondot, Jan 17 2016
It is a plausible conjecture that there are no more terms, but this has not been proved. - N. J. A. Sloane, Nov 17 2017

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

cf. A146025, A146027, A131646, A275600

Select[Range[0, 10^5], Function[n, Times @@ Boole@ Map[Max@ IntegerDigits[n, #] <= 3 &, Range[2, 9]] > 0]] (* Michael De Vlieger, Aug 15 2016 *)

A275600 Numbers that can be written in all bases from base 2 to base 6 using only the digits 0, 1 and 2.

Original entry on oeis.org

0, 1, 2, 6, 36, 37, 260, 1302, 1376, 1380, 1381, 1382, 1556, 1560, 1561, 1562, 16932, 562500, 562501, 562502, 562506, 562512, 562536, 562537, 562752, 562760, 23610752, 23610756, 23610757, 23610786, 23615750, 23615760, 23615761, 23615762, 23615785, 23615786, 23626310

Offset: 1

Sergio Pimentel, Aug 03 2016

Is there any number that keeps this property also in base 7, other than the trivial cases 0,1,2?

			16932 is in the sequence because this number can be written in bases 2 through 6 using only the digits 0, 1 and 2: 16932(b4)  = 10020210 / (b5) = 1020212 / (b6) = 210220.

Rémy Sigrist and Chai Wah Wu, Table of n, a(n) for n = 1..10000 [Terms 1 through 187 by Chai Wah Wu]

Cf. A146025, A146026, A146027, A146028, A146029, A258107, A258981.

Select[Range[10^6], Function[k, Max@ Flatten@ Map[IntegerDigits[k, #] &, Range[4, 6]] < 3]] (* or *)
Select[Range[10^5], Function[k, Total@ Flatten@ Map[Take[RotateRight@ DigitCount[k, #], -(# - 3)] &, Range[4, 6]] == 0]] (* (not as efficient) Michael De Vlieger, Aug 03 2016 *)

nextWithSmallDigits(n, base) = my (pow=1, rem=n, val=0, d); while (rem>0, d = rem % base; rem = rem \ base; if (d>2, val = 0; rem = rem+1, val = val + d*pow); pow = pow * base); return (val)
{ n = 0; prev = 0; while (n < 300, succ = prev; for (b=4,6, succ = nextWithSmallDigits(succ, b)); if (prev==succ, n = n+1; print(n " " prev); prev = succ+1, prev = succ)) } \\ Rémy Sigrist, Sep 08 2016

use ntheory ":all"; my($x,$n10)=(0,0); while ($x < 50) { my $n = fromdigits( todigitstring($n10++, 3), 6);  next if vecany { $ > 2 } todigits($n, 4);  next if vecany { $ > 2 } todigits($n, 5);  print ++$x," $n\n"; } # Dana Jacobsen, Aug 16 2016

from gmpy2 import digits
A275600_list = [n for n in (int(digits(m,3),6) for m in range(10**6)) if max(digits(n,5)) <= '2' and max(digits(n,4)) <= '2'] # Chai Wah Wu, Aug 15 2016

a(18)-a(26) from Michael De Vlieger, Aug 03 2016

a(27)-a(37) from Chai Wah Wu, Aug 15 2016

A131646 Numbers that can be written from base 2 to base 18 using only the digits 0 to 9 (conjectured to be complete).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 18, 19, 20, 1027, 1028, 1029, 14745, 9020076688681, 9439828025162228377, 9439829801208141318

Offset: 1

Daniel Mondot, Sep 08 2007, Nov 02 2008

Originally checked to 2^20356 (or 5.8*10^6127) in Nov 2008.
It appears that 19 and 20 are the only numbers > 9 that can be written up to base 19 only using digits 0 to 9 and 20 is the only number > 9 that can be written up to base 20 only using digits 0 to 9.
It is a plausible conjecture that there are no more terms, but this has not been proved. - N. J. A. Sloane, Nov 17 2017

B. R. Barwell, Numbers Without Letters, Journal of Recreational Mathematics, Vol. 25:3 (1993), 174-179.

Daniel Mondot, Related Puzzle.

Cf. A146025, A146026, A146027, A146028, A146029.

f[n_] := Total[Total@ Drop[RotateRight[DigitCount[n, #]], 10] & /@ Range[11, 18]]; Select[Range[0, 20000], f@ # == 0 &] (* Michael De Vlieger, Aug 29 2015 *)

isok(n) = if (n, for (b=11, 18, if (vecmax(digits(n,b))>9, return(0)))); 1; \\ Michel Marcus, Aug 30 2015

Edited by Charles R Greathouse IV, Nov 01 2009

Reference added by William Rex Marshall, Oct 23 2011

A146028 Numbers that can be written from base 2 to base 15 using only the digits 0 to 7.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 15, 16, 174731235562130, 174731235562131, 174731235562132, 174731235562143, 174731235562147, 174731235562170, 174731235562171, 174731235564710, 174731235564711, 174731236371006, 25354527232277132536350, 25354527232277132536351

Offset: 1

Daniel Mondot, Nov 02 2008, Nov 06 2008

It is a plausible conjecture that there are no more terms, but this has not been proved. - N. J. A. Sloane, Nov 17 2017

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

Cf. A146025, A146026, A146027, A146029, A131646, A275600.

Select[Range[0, 10^5], Function[n, Times @@ Boole@ Map[Max@ IntegerDigits[n, #] <= 7 &, Range[2, 15]] > 0]] (* Michael De Vlieger, Aug 15 2016 *)

A146029 Numbers that can be written from base 2 to base 17 using only the digits 0 to 8 (conjectured to be complete).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 17, 18

Offset: 1

Daniel Mondot, Nov 02 2008

It is a plausible conjecture that there are no more terms, but this has not been proved. - N. J. A. Sloane, Nov 17 2017

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

Cf. A146025, A146026, A146027, A146028, A131646, A275600.

Select[Range[0, 10^5], Function[n, Times @@ Boole@ Map[Max@ IntegerDigits[n, #] <= 8 &, Range[2, 17]] > 0]] (* Michael De Vlieger, Aug 15 2016 *)

cp's OEIS Frontend

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

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A146026 Numbers that can be written from base 2 to base 9 using only the digits 0 to 3.

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A275600 Numbers that can be written in all bases from base 2 to base 6 using only the digits 0, 1 and 2.

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A131646 Numbers that can be written from base 2 to base 18 using only the digits 0 to 9 (conjectured to be complete).

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A146028 Numbers that can be written from base 2 to base 15 using only the digits 0 to 7.

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A146029 Numbers that can be written from base 2 to base 17 using only the digits 0 to 8 (conjectured to be complete).

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