A146027 Numbers that can be written from base 2 to base 10 using only the digits 0 to 4.

0, 1, 2, 3, 4, 10, 100, 140004, 140304, 140312, 1131032, 1131033, 1131034, 1131040

Offset: 1

Author

Daniel Mondot, Oct 26 2008

Keywords

• nonn
• base

Comments

Conjectured to be complete.

Checked on Oct 31 2008 up to 2^16384 (or 1.2*10^4932). - Daniel Mondot, Jan 17 2016

No more entries < 10^78. - Robert Israel, Aug 31 2015

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

Links

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

Crossrefs

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

Programs

Maple

imax:= 20: # to consider numbers < 6^imax
L:= Matrix(5,imax):
Delta:= proc(L,b)
  local i,j,m,Lloc;
  if max(L) <= 4 then return 0 fi;
  Lloc:= L;
  m:= 0;
  for j from 1 to imax while max(Lloc[j..imax]) > 4 do
          m:= m + b^(j-1)*(b-Lloc[j]);
          if j < imax then Lloc[j+1]:= Lloc[j+1]+1 fi
  od;
  m
end proc:
n:= 0: count:= 1: A[1]:= 0:
isdone:= false;
while max(L[..,imax]) < 5 and not isdone do
  n:= n+1;
  L[..,1]:= L[..,1]+<1,1,1,1,1>;
  m:= max(seq(Delta(L[b-5,..],b),b=6..10));
  while m > 0 and not isdone do
    n:= n+m;
    for b from 6 to 10 do
       Lb:= convert(n,base,b);
       if nops(Lb) > imax then isdone:= true; break fi;
       L[b-5,1..nops(Lb)]:= Vector[row](Lb);
    od:
    m:= max(seq(Delta(L[b-5,..],b),b=6..10));
  od;
  if not isdone then
    count:= count+1;
    A[count]:= n;
  fi
od:
seq(A[i],i=1..count); # Robert Israel, Aug 31 2015

Mathematica

f[n_] := Total[Total@ Drop[RotateRight[DigitCount[n, #]], 5] & /@ Range[6, 10]]; Select[Range[0, 1200000], f@ # == 0 &] (* Aug 29 2015, or *)
Select[Range[0, 1200000], Function[n, Times @@ Boole@ Map[Max@ IntegerDigits[n, #] <= 4 &, Range[2, 10]] > 0]] (* Michael De Vlieger, Aug 15 2016 *)

PARI

isok(n) = if (n, for (b=6, 10, if (vecmax(digits(n,b))>4, return(0)))); 1; \\ Michel Marcus, Aug 30 2015

Extensions

Edited by Charles R Greathouse IV, Nov 01 2009