Sum of $n$ Consecutive Positive Integers Divided by $n$ and $(n+1)$

• Case-A : $1$ is divisible by $\color{#20A900} 1$ , but $\color{#D61F06} \text{not}$ by $2=({\color{#20A900} 1}+1)$ .

• Case-B : $2+3$ is $\color{#D61F06} \text{neither}$ divisible by $\color{#20A900} 2$ , $\color{#D61F06} \text{nor}$ by $3=({\color{#20A900} 2}+1)$ .

• Case-C : $3+4+5$ is divisible by both $\color{#20A900} 3$ and $4=({\color{#20A900} 3}+1)$ .

Case-A describes a representative sum of $n$ consecutive positive integers that is divisible by $n$ , but $\color{#D61F06} \text{not}$ by $(n+1)$ .

Case-B describes a representative sum of $n$ consecutive positive integers that is $\color{#D61F06} \text{neither}$ divisible by $n$ , $\color{#D61F06} \text{nor}$ by $(n+1)$ .

Case-C describes a representative sum of $n$ consecutive positive integers that is divisible by both $n$ and $(n+1)$ .

Now, Let $S_n$ be a sequence of $n$ consecutive positive integers and $p$ be its least element. Furthermore, the sum of integers in $S_n$ is $\color{#D61F06} \text{not}$ divisible by $n$ , but divisible by $(n+1)$ . Also, let $F(n)$ denote the number of such sequence(s) $S_n$ with $p \leq 2017$ .

Find $F(2)+F(31)$ .