This is a good question.
By experiment, 1 is not, 2 is not, 4 is not, 8 is not, 16 is not. So there are some counting numbers that are not the sum of two or more counting numbers.
It turns out that a counting number can be expressed as the sum of two or more consecutive counting numbers if and only if it is not a power of 2.
Let us do some algebra (sorry about the lack of nice symbols).
Consider n consecutive counting numbers, starting at a (so n >= 2, a >= 1).
The numbers are a, a+1, ... , a + (n-1).
If we add these up, we get the sum s = na + n(n-1)/2.
For example, five consecutive numbers starting at 7 are 7, 8, 9, 10, 11.
Here n = 5 and a = 7. The sum of the five numbers is s = 5x7 + 5x(5-1)/2 = 45.
Fact: Every sum of at least two consecutive counting numbers has an odd factor.
Proof: We analyse s = na + n(n-1)/2. There are two cases.
Case 1: n is odd. Then s = n(a + (n-1)/2). Since n is odd, n-1 is even, so (n-1)/2 is a whole number, and a + (n-1)/2 is also a whole number.
Then s has a factor of n, and n is odd.
Case 2: n is even. Write m = n/2, so m is a whole number.
We find s = 2ma + 2m(2m-1)/2. so
s = 2ma + m(2m-1).
Thus s = m(2a + 2m - 1).
Whatever a and m are, 2a + 2m must be even, so 2a + 2m - 1 is odd.
Thus s has an odd factor, namely 2a + 2m - 1.
Consequence: If a counting number is a power of 2, it cannot be expressed as the sum of two or more consecutive counting numbers, because a number that is a power of 2 does not have any odd factors.
That is half the story. As far as the Fact above tells us, there might be some numbers that do have odd factors and cannot be expressed as the sum of two or more consecutive counting numbers.
Actually, there are not.
Any counting number can be written as a product pq, where p is a power of 2, and q is an odd number (which could be 1). If the number has an odd factor, q is at least 3.
For example, 80 = 16 x 5, so p = 16, q = 5 for the number 80.
Fact: Every counting number that has an odd factor can be expressed as the sum of two or more consecutive counting numbers.
Proof: Write our counting number k as the product pq; since k is supposed to have an odd factor, q is at least 3. We note that 2p = q is impossible, because 2p is even and q is odd. There are two cases.
Case 1: 2p < q.
Set n = 2p and a = (q - 2p + 1)/2.
Since 2p < q, q - 2p + 1 is positive, and since q is odd, q - 2p + 1 is even. So a = (q - 2p + 1)/2 is a counting number.
Let us work out s = na + n(n-1)/2. We get
s = (2p)(q - 2p + 1)/2 + (2p)(2p - 1)/2
So s = p(q - 2p + 1) + p(2p-1), which works out to pq, as it should.
Thus in Case 1 k = pq is the sum of nconsecutive counting numbers starting at a.
Case 2: 2p > q.
Set n = q and a = p - (q-1)/2.
Since 2p > q, certainly 2p > q - 1, so 2p - (q - 1) is positive. Also, since q is odd, p - (q - 1) is even. So a = p - (q-1)/2 is a counting number.
Let us work out s = na + n(n-1)/2. We get
s = q( p - (q-1)/2) + q(q-1)/2. This works out to pq.
Thus in Case 2 k = pq is the sum of nconsecutive counting numbers starting at a.
In both cases, k = pq is the sum of nconsecutive counting numbers starting at a, and our proof is finished.
For an example, try k = 68. Since 68 = 4 x 17, p = 4 and q = 17. Since 2 x 4 < 17, this is an example of Case 1.
Set n = 2p = 8, and a = (q - 2p + 1)/2 = 5. Then 68 should be the sum of 8 consecutive numbers starting at 5.
Check: 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 = 68.
For another example, try k = 80. We have 80 = 16 x 5, so p = 16, q = 5. Since 2 x 16 > 5, this is an example of Case 2.
Set n = q = 5 and a = p - (q-1)/2 = 14. Then 80 should be the sum of 5 consecutive numbers starting at 14.
Check: 14 + 15 + 16 + 17 + 18 = 80.
The two Facts above give us a complete solution to the problem: a counting number can be expressed as the sum of two or more consecutive counting numbers if and only if it is not a power of 2.
Comment: We have shown that a number that has an odd factor can be expressed as the sum of two or more consecutive counting numbers, and the proof of the second Fact shows a way to do it. But there may be more than one way of expressing a number as the sum of two or more consecutive counting numbers.
For example, 60 = 4 x 15, so it fits under Case 1, and we find that n = 8, and a = 4.
So 60 = 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11.
But also 60 = 10 + 11 + 12 + 13 + 14, which doesn't come out of our proof.
triangular numbers are created when all numbers are added for example: To find the 5 triangular number (1+2+3+4+5).
No- not exactly. Negative integers are not counting numbers. Positive integers are identified with counting numbers. Many authors like to start with zero as a counting number.
The numbers 2 and 3 are consecutive prime numbers. Are there other pairs of prime numbers which are consecutive numbers?
Usually all the integers (counting numbers) from 1 to 100.Usually all the integers (counting numbers) from 1 to 100.Usually all the integers (counting numbers) from 1 to 100.Usually all the integers (counting numbers) from 1 to 100.
Yes all counting numbers are whole numbers, but the reverse is not true (zero!)
They are all next to eachother when counting normally example: 1,2,3 or, 23,24,25,26 When doing certain math equasions, you will also use consecutive numbers. In that case, they are special becuse they all add up to the number and they are all next to eachother when counting normally
They are all whole numbers under ten; consecutive counting numbers.
triangular numbers are created when all numbers are added for example: To find the 5 triangular number (1+2+3+4+5).
No- not exactly. Negative integers are not counting numbers. Positive integers are identified with counting numbers. Many authors like to start with zero as a counting number.
The numbers 2 and 3 are consecutive prime numbers. Are there other pairs of prime numbers which are consecutive numbers?
Yes all counting numbers are whole numbers, but the reverse is not true (zero!)
Yes.all counting numbers Have factor.
Negative numbers are not counting numbers. Counting numbers are the integers starting with 1 and then 2 and so forth.
Usually all the integers (counting numbers) from 1 to 100.Usually all the integers (counting numbers) from 1 to 100.Usually all the integers (counting numbers) from 1 to 100.Usually all the integers (counting numbers) from 1 to 100.
Yes all counting numbers are whole numbers, but the reverse is not true (zero!)
no
2 and 3 are the only consecutive numbers that are prime.