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

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 20, 29

Offset: 1

Author

Patrick De Geest, Jan 15 2000

Keywords

• nonn
• base
• hard
• more

Comments

Previous name was "Smallest power of 2 whose digits occur with same frequency n".

Next term > 3597.

Not multiplicative since a(18) is supposedly > 3597, but a(2) = 2 and a(9) = 9. - David W. Wilson, Jun 12 2005

From Robert Israel, Aug 14 2015: (Start)

Next term (if any) > 10^4.

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)

Examples

			E.g., 2^29 = 536870912 where each digit occurs once in this case.
		

Crossrefs

Cf. A052069, A052070, A052071, A052072.

Programs

Maple

filter:= proc(n) local x,i,P;
P:= add(x^i,i=convert(2^n,base,10));
nops({coeffs(P,x)})=1
end proc:
select(filter, [$1..10^4]); # Robert Israel, Aug 14 2015

Extensions

Name and offset corrected by Michel Marcus, Aug 12 2015