cp's OEIS Frontend

A052060 Numbers n such that the digits of 2^n occur with the same frequency.

%I A052060 #31 Aug 14 2015 15:08:15
%S A052060 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,20,29
%N A052060 Numbers n such that the digits of 2^n occur with the same frequency.
%C A052060 Previous name was "Smallest power of 2 whose digits occur with same frequency n".
%C A052060 Next term > 3597.
%C A052060 Not multiplicative since a(18) is supposedly > 3597, but a(2) = 2 and a(9) = 9. - _David W. Wilson_, Jun 12 2005
%C A052060 From _Robert Israel_, Aug 14 2015: (Start)
%C A052060 Next term (if any) > 10^4.
%C A052060 It is highly likely that the sequence is finite.  For each m, there are at most 4 powers of 2 with m digits.  If m is large, of the 9*10^m numbers with m digits, there are at most about c * 10^m/m^(9/2) with equal digit frequencies where c is a constant (this comes from the case where there all 10 digits are represented with frequencies m/10).   Thus heuristically the expected number of m-digit powers of 2 with equal digit frequencies decreases like m^(-9/2), and this has a finite sum. (End)
%e A052060 E.g., 2^29 = 536870912 where each digit occurs once in this case.
%p A052060 filter:= proc(n) local x,i,P;
%p A052060 P:= add(x^i,i=convert(2^n,base,10));
%p A052060 nops({coeffs(P,x)})=1
%p A052060 end proc:
%p A052060 select(filter, [$1..10^4]); # _Robert Israel_, Aug 14 2015
%Y A052060 Cf. A052069, A052070, A052071, A052072.
%K A052060 nonn,base,hard,more
%O A052060 1,3
%A A052060 _Patrick De Geest_, Jan 15 2000
%E A052060 Name and offset corrected by _Michel Marcus_, Aug 12 2015