, 2019

We consider first the integers that can be expressed as the sum of exactly 4 consecutive positive integers.

The smallest such integer is $1+2+3+4=10$. The next smallest such integer is $2+3+4+5=14$.

We note that when we move from $k+(k+1)+(k+2)+(k+3)$ to $(k+1)+(k+2)+(k+3)+(k+4)$, we add 4 to the total (this equals the difference between $k+4$ and $k$ since the other three terms do not change).

Therefore, the positive integers that can be expressed as the sum of exactly 4 consecutive positive integers are those integers in the arithemetic sequence with first term 10 and common difference 4.

Since $n \leq 100$, these integers are $$10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98$$ There are 23 such integers.

Next, we consider the positive integers $n \leq 100$ that can be expressed as the sum of exactly 5 consecutive positive integers.

The smallest such integer is $1+2+3+4+5=15$ and the next is $2+3+4+5+6=20$.

Using an argument similar to that from above, these integers form an arithemetic sequence with first term 15 and common difference $5$.

Since $n \leq 100$, these integers are $15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100$.

When we exclude the integers already listed above (30, 50, 70, 90), we obtain $$15, 20, 25, 35, 40, 45, 55, 60, 65, 75, 80, 85, 95, 100$$ There are 14 such integers.

Next, we consider the positive integers $n \leq 100$ that can be expressed as the sum of exactly 6 consecutive positive integers.

These integers form an arithmetic sequence with first term 21 and common difference 6.

Since $n \leq 100$, these integers are $21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99$.

When we exclude the integers already listed above (45, 75), we obtain $$21, 27, 33, 39, 51, 57, 63, 69, 81, 87, 93, 99$$ There are 12 such integers.

Since $1+2+3+4+5+6+7+8+9+10+11+12+13+14 = 105$ and this is the smallest integer that can be expressed as the sum of 14 consecutive positive integers, then no $n \leq 100$ is the sum of 14 or more consecutive positive integers. (Any sum of 15 or more consecutive positive integers will be larger than 105.)

Therefore, if an integer $n \leq 100$ can be expressed as the sum of $s \geq 4$ consecutive integers, then $s \leq 13$.

We make a table to enumerate the $n \leq 100$ that come from values of $s$ with $7 \leq s \leq 13$ that we have not yet counted:

$s$
$n$
$n = 100$
New $n$

7
28
28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98
28, 49, 56, 77, 84, 91

8
36
36, 44, 52, 60, 68, 76, 84, 92, 100
36, 44, 52, 68, 76, 92

9
45
45, 54, 63, 72, 81, 90, 99
72

10
55
55, 65, 75, 85, 95
None

11
66
66, 77, 88, 99
88

12
78
78, 90
None

13
91
91
None

In total, there are $23+14+12+6+6+1+1=63$ such $n$.

What do you notice about the $n$ that cannot expressed in this way?