Any positive integer (decimal) can be represented as the corresponding binary,

For example, the binary representation of 100 is 1 100 100.

The fifth power of 2 is expressed as 100000, plus five zeros.

The integer symbol [x] in the title is constantly deprived of value.

Below we all use binary numbers to analyze. take notice of

Let the exponent of a denominator be k, and items with no more than k digits will become decimals in the result and be deleted.

That's how things developed,

Ok, so the smallest integer is required, and its expression must be * * *11111.

Move the decimal point of binary (the initial value is on the far right) to the left gradually, followed by the remainder, which is also a binary representation.

Then the remaining total is 0.5+(0.5+0.25)+ ...

Each term is a geometric series, which is the sum of series.

The general term is 1- 1/2 n,

So its sum is n- 1+ 1/2 n,

Don't forget that the margin on the graph is 1- 1/2 n (that number is exactly 1 less than 2 n).

So the total profit is n,

When n=9, the number of binary digits is111111,* * 9 1, and the number is.