What is the 50th smallest positive integer that can be written as the sum of distinct non-negative integer powers of 3?
from briliant
perhaps it's 327
To find the 50th smallest positive integer that can be written as the sum of distinct non-negative integer powers of 3, let's start by listing some of these numbers and see if we can find a pattern:
1 (3^0)
3 (3^1)
4 (3^0 + 3^1)
9 (3^2)
10 (3^0 + 3^2)
12 (3^1 + 3^2)
...
(continuing the pattern)
By observing the numbers listed above, we can see that they can be created by choosing a combination of distinct non-negative integer powers of 3.
To find the 50th smallest positive integer, we need to systematically generate these numbers until we reach the 50th one.
One approach to generate these numbers is to use a binary representation of the numbers from 1 to 3^k - 1, where k is the maximum power of 3 allowed in the sum.
For example, for k = 3 (maximum power of 3 allowed is 3^3 = 27), we can generate the binary representation of the numbers from 1 to 26 (3^k - 1):
0001 (1)
0010 (2)
0011 (3)
0100 (4)
0101 (5)
...
(continuing until 11010 (26))
Then, using this binary representation, we can convert it into the sum of distinct non-negative integer powers of 3 by multiplying each bit by the corresponding power of 3.
For example, for the binary representation 0010 (2), we would multiply the first bit by 3^1 and the second bit by 3^2, resulting in the number 2 * 3^1 + 0 * 3^2 = 6.
By generating these numbers in increasing order and checking if they are distinct and positive, we can find the 50th smallest positive integer that can be written as the sum of distinct non-negative integer powers of 3.