A263684 - OEIS

%I #21 Jul 30 2022 20:09:36

%S 0,1,5,16400,16401,16405,82000,82001,82005

%N Numbers whose base-4 and base-5 representations have only 0's and 1's.

%C Intersection of A000695 and A033042.

%C These appear to be all the terms. There are no more below 10^500.

%e 16400 is 10000100 in base 4 and 1011100 in base 5.

%p split:= proc(ab, B)

%p    local a,b,La, Lb, k, j, a1,  a2, b1, b2, x;

%p    global Res, count;

%p    a:= ab[1]; b:= ab[2];

%p    if b-a <= 1000 then

%p       for x from a to b-1 do

%p         if max(convert(x,base,4)) <= 1 and max(convert(x,base,5)) <= 1 then

%p            count:= count+1; Res[count]:= x

%p         fi

%p       od;

%p       return ({});

%p    fi;

%p    La:= convert(a,base,B);

%p    Lb:= convert(b,base,B);

%p    if nops(Lb) > nops(La) then La:= [op(La),0$(nops(Lb)-nops(La))] fi;

%p    k:= ListTools:-SelectLast(`>`,Lb-La,0,output=indices);

%p    if La[k] = 0 then

%p      a1:= a;

%p      b1:= 2 + add(B^i,i=1..k-2) + add(La[i]*B^(i-1),i=k+1..nops(La));

%p      a2:= B^(k-1) + add(La[i]*B^(i-1),i=k+1..nops(La));

%p      b2:= min(b, b1 + B^(k-1));

%p      return(select(t -> (t[1]<t[2]),{[a1,b1],[a2,b2]}));

%p    elif La[k] = 1 then

%p      a1:= a;

%p      b1:= 2 + add(B^i,i=1..k-2) + add(La[i]*B^(i-1),i=k..nops(La));

%p      return(select(t -> t[1]<t[2], [[a1,b1]]))

%p    else return ( {})

%p    fi;

%p end proc:

%p count:= 0: Res:= 'Res':

%p Cands:= {[0,10^400]}:

%p while nops(Cands) > 0 do

%p    Cands:= map(op@split, Cands, 5);

%p    Cands:= map(op@split, Cands, 4);

%p od:

%p sort(convert(Res,list));

%t Select[Range[0,83000],Max[Join[IntegerDigits[#,4],IntegerDigits[#,5]]]<2&] (* _Harvey P. Dale_, Sep 04 2018 *)

%o (PARI) isok(n) = (n==0) || ((vecmax(digits(n,4))<=1) && (vecmax(digits(n,5))<=1)); \\ _Michel Marcus_, Oct 24 2015

%Y Cf. A000695, A033042, A146025, A258981.

%K nonn,fini,full

%O 1,3

%A _Robert Israel_, Oct 23 2015