cp's OEIS Frontend

A268337 Numbers which have only digits 0 and 1 in bases 3 and 5.

%I A268337 #27 Dec 13 2021 15:37:49
%S A268337 0,1,30,31,756,3250,3276,3280,81255,81256,81280,81900,81901,82000,
%T A268337 59078250,59078251,59078280,59078281,31789468750,31789468776,
%U A268337 31789469505,31789469506,31789471900,31789471905,31789471906,31789472005,946095722031,946095800025,946095800026,946095800031,946095800130
%N A268337 Numbers which have only digits 0 and 1 in bases 3 and 5.
%C A268337 The number 82000 is famous for having only digits 0 and 1 in all bases <= 5, no other such number > 1 is known. See also A146025 and A258981.
%C A268337 If explicit formulas for (convenient) infinite subsequences of this one can be found, this could open new ways to progress on this problem.
%C A268337 The terms come in groups having roughly the first half (or at least third) of digits in common, see the link "Terms in base 10, 5 and 3".
%H A268337 Ray Chandler, <a href="/A268337/b268337.txt">Table of n, a(n) for n = 1..10000</a> (first 84 terms from M. F. Hasler, next 31 terms from Charles R Greathouse IV)
%H A268337 M. F. Hasler, <a href="/A268337/a268337.pdf">Terms in base 10, 5 and 3.</a>
%F A268337 a(n) >> n^k with k = log 5/log 2 = 2.321928.... - _Charles R Greathouse IV_, Feb 02 2016
%p A268337 d:= 20: # to get all terms < 5^d
%p A268337 res:= NULL:
%p A268337 T:= combinat:-cartprod([[$0..1]$d]):
%p A268337 while not T[finished] do
%p A268337   r:= T[nextvalue]();
%p A268337   v:= add(r[i]*5^(d-i),i=1..d);
%p A268337   if max(convert(v,base,3)) <= 1 then
%p A268337     res:= res,v
%p A268337   fi
%p A268337 od:
%p A268337 res; # _Robert Israel_, Feb 01 2016
%t A268337 Module[{t=Tuples[{0,1},25],b3,b5},b3=FromDigits[#,3]&/@t;b5=FromDigits[ #,5]&/@t;Intersection[b3,b5]] (* The program generates the first 26 terms of the sequence. *) (* _Harvey P. Dale_, Dec 13 2021 *)
%o A268337 (PARI) print1(0);for(n=1,1e10,vecmax(digits(t=subst(Pol(binary(n)),'x,5),3))<2&&print1(","t))
%o A268337 (PARI) list(lim)=my(v=List([0]),d=digits(lim\1,5),t); for(i=1,#d, if(d[i]>1, for(j=i,#d, d[j]=1); break)); for(n=1,fromdigits(d,5), t=fromdigits(binary(n),5); if(vecmax(digits(t,3))<2, listput(v,t))); Vec(v) \\ _Charles R Greathouse IV_, Feb 02 2016
%Y A268337 Cf. A005836, A033042.
%K A268337 nonn,base
%O A268337 1,3
%A A268337 _M. F. Hasler_, Feb 01 2016