|
%I #31 Aug 14 2015 15:08:15
%S 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,20,29
%N Numbers n such that the digits of 2^n occur with the same frequency.
%C Previous name was "Smallest power of 2 whose digits occur with same frequency n".
%C Next term > 3597.
%C Not multiplicative since a(18) is supposedly > 3597, but a(2) = 2 and a(9) = 9. - _David W. Wilson_, Jun 12 2005
%C From _Robert Israel_, Aug 14 2015: (Start)
%C Next term (if any) > 10^4.
%C 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 E.g., 2^29 = 536870912 where each digit occurs once in this case.
%p filter:= proc(n) local x,i,P;
%p P:= add(x^i,i=convert(2^n,base,10));
%p nops({coeffs(P,x)})=1
%p end proc:
%p select(filter, [$1..10^4]); # _Robert Israel_, Aug 14 2015
%Y Cf. A052069, A052070, A052071, A052072.
%K nonn,base,hard,more
%O 1,3
%A _Patrick De Geest_, Jan 15 2000
%E Name and offset corrected by _Michel Marcus_, Aug 12 2015
|
