# Sum of power of two exact 2010 different ways

• Christian1992
In summary: I'm not sure if you can do this or not. Hello,In summary, the person does not know how to find a solution to the problem.
Christian1992
Hello,

I look for a solution for the following problen.

Determine all numbers, which can be written in exact 2010 different ways as a sum of powers of two with non-negative exponent, while all exponents are only allowed to appear maximal three times in one sum.
(All sums where only the order of the summands is changed are count as one sum.)

My problem is that I do not have any idea, how to find a solution to this task.
Therefore I started to determine in how many different sums I can write for example 2^4:
1. 2^4
2. 2*2^3
3. 2*2^2+2^3
4. 2*2+2^2+2^3
5. 2*2^0+2+2^2+2^3
6. 2*2^0+3*2+2^3
7. 2*2^0+3*2+2*2^3

Nevertheless, I did not find any coherenz zu the number and the number of different sums.

Do you have any tips?

Christian

No idea?

Christian1992 said:
Hello,

I look for a solution for the following problen.

Determine all numbers, which can be written in exact 2010 different ways as a sum of powers of two with non-negative exponent, while all exponents are only allowed to appear maximal three times in one sum.
(All sums where only the order of the summands is changed are count as one sum.)
I don't understand what you're asking. Is it that you want all numbers that can be written in exactly 2010 different ways?

Or is it that you want to find the number of ways that 2010 can be written as a sum of powers of 2?
Christian1992 said:
My problem is that I do not have any idea, how to find a solution to this task.
Therefore I started to determine in how many different sums I can write for example 2^4:
1. 2^4
2. 2*2^3
3. 2*2^2+2^3
4. 2*2+2^2+2^3
5. 2*2^0+2+2^2+2^3
6. 2*2^0+3*2+2^3
7. 2*2^0+3*2+2*2^3

Nevertheless, I did not find any coherenz zu the number and the number of different sums.

Do you have any tips?

Christian

Christian1992 said:
Hello,

I look for a solution for the following problen.

Determine all numbers, which can be written in exact 2010 different ways as a sum of powers of two with non-negative exponent, while all exponents are only allowed to appear maximal three times in one sum.
(All sums where only the order of the summands is changed are count as one sum.)

My problem is that I do not have any idea, how to find a solution to this task.
Therefore I started to determine in how many different sums I can write for example 2^4:
1. 2^4
2. 2*2^3
3. 2*2^2+2^3
4. 2*2+2^2+2^3
5. 2*2^0+2+2^2+2^3
6. 2*2^0+3*2+2^3
7. 2*2^0+3*2+2*2^3

Nevertheless, I did not find any coherenz zu the number and the number of different sums.

Do you have any tips?

Christian

Hi there!
I believe you quote the question from Columbus state university .

This is an easy question.
Assume there is four blocks since the two non zero integer are similar ,
we label one of the a and b. one with a selection of 3 integers another with 4.
For a>2.

Therefore the blocks of 4 integer will look like this:
a00b
a0b0
ab00

I'll leave the rest to you.
ans is thirty-_____ .

icystrike said:
Hi there!
I believe you quote the question from Columbus state university .

This is an easy question.
Assume there is four blocks since the two non zero integer are similar ,
we label one of the a and b. one with a selection of 3 integers another with 4.
For a>2.

Therefore the blocks of 4 integer will look like this:
a00b
a0b0
ab00

I'll leave the rest to you.
ans is thirty-_____ .

Huh? You're sure this is the answer to the right question?

Christian1992 said:
Hello,

I look for a solution for the following problen.

Determine all numbers, which can be written in exact 2010 different ways as a sum of powers of two with non-negative exponent, while all exponents are only allowed to appear maximal three times in one sum.
(All sums where only the order of the summands is changed are count as one sum.)

My problem is that I do not have any idea, how to find a solution to this task.
Therefore I started to determine in how many different sums I can write for example 2^4:
1. 2^4
2. 2*2^3
3. 2*2^2+2^3
4. 2*2+2^2+2^3
5. 2*2^0+2+2^2+2^3
6. 2*2^0+3*2+2^3
7. 2*2^0+3*2+2*2^3

Nevertheless, I did not find any coherenz zu the number and the number of different sums.

there are actually 9 ways to write 2^4

writing them in 'binary'

10000
2000
1200
1120
1112
1032
320 = 3*2^2 + 2 * 2^2
312 = 3*2^2 + 1 * 2^1 + 2*2^0
232 = 2*2^2 + 3 * 2^1 + 2*2^0

you missed the last 3 and your number 7 isn't valid.

It appears that if f(n) is the number of ways to write n, then f(n) = 1 + floor(n/2)

(floor(x) is the greatest integer <= x)

I haven't been able to prove this so far. It's easy to prove that f(2n+1) = f(2n) so you
only have to prove f(n) = 1+n/2 for even numbers.

## 1. What is the concept of "Sum of power of two exact 2010 different ways"?

The concept refers to finding all possible ways to express the number 2010 as the sum of different powers of two. For example, 2010 can be expressed as 2^10 + 2^7 + 2^4 + 2^1.

## 2. Why is this concept important in science?

This concept is important in fields such as computer science and cryptography, where numbers are often represented in binary form using powers of two. Understanding how to break down a number into different powers of two can be useful in solving certain problems or creating more efficient algorithms.

## 3. What is the significance of the number 2010 in this concept?

The number 2010 is significant because it is the smallest number that can be expressed as the sum of exactly 2010 different powers of two. This makes it a unique and interesting number to study in relation to this concept.

## 4. How many ways are there to express 2010 as the sum of different powers of two?

There are exactly 2010 different ways to express 2010 as the sum of different powers of two. This has been proven mathematically through various methods, including dynamic programming and combinatorics.

## 5. Can this concept be extended to other numbers besides 2010?

Yes, this concept can be extended to any positive integer. However, the number of ways to express a given number as the sum of different powers of two will vary depending on the number itself. For example, 2010 has 2010 different ways, while 10 only has 5 different ways.

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