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LSpice
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A deduction about the The asymptotic of $|\{1\leq n\leq x|\gcd(n,S(n))=1\}|$, with $S(n)$ the sum of remainders, and get idea for other miscellany problem

Let $n\geq 1$ be an integer, in. In this post we denote the sum of remainders function as $$S(n)=\sum_{k=1}^n n\text{ mod }k,$$$$S(n)=\sum_{k=1}^n n \bmod k,$$ for example $S(1)=S(2)=0+0$ and $S(5)=0+1+2+1+0=4$. In the literature there are problems that were studied related to the condition $\gcd(n,f(n))=1$, for a given arithmetic function $f(n)$.

Question. 

A) Is it possible to provide roughly a cheap bound for the cardinality $$\#\{1\leq n\leq x|\gcd(n,S(n))=1\}$$ as $x$ grows to $\infty$? 

B) The sequence of primes $p$ such that satisfy the condition $$\gcd(p,S(p))>1$$ starts as $2,11,17,2161,\ldots$. Can you provide us any idea about if this sequence havehas finitely many terms? Many thanks.

Just to emphasize, since I'm asking two questions, only is required that you provide a cheap bound for A) and a suitable reasoning/heuristic for B), to get idea for these problems.

Computational evidence and documentation for Question B. We've the following script in Pari/GP showing the first few terms

for(n=1, 10000, if(gcd(n,sum(k=1,n,n%k))>1&&isprime(n)==1,print(n)))

that you can evaluate on the website Sage Cell ServerSage Cell Server choosing as Language GP. Here the string sum(k=1,n,n%k) is our sum of remainders $S(n)$ with n%k coding $n\text{ mod }k$$n \bmod k$ for each integer $1\leq k\leq n$.

A deduction about the asymptotic of $|\{1\leq n\leq x|\gcd(n,S(n))=1\}|$, with $S(n)$ the sum of remainders, and get idea for other miscellany problem

Let $n\geq 1$ be an integer, in this post we denote the sum of remainders function as $$S(n)=\sum_{k=1}^n n\text{ mod }k,$$ for example $S(1)=S(2)=0+0$ and $S(5)=0+1+2+1+0=4$. In the literature there are problems that were studied related to the condition $\gcd(n,f(n))=1$, for a given arithmetic function $f(n)$.

Question. A) Is it possible to provide roughly a cheap bound for the cardinality $$\#\{1\leq n\leq x|\gcd(n,S(n))=1\}$$ as $x$ grows to $\infty$? B) The sequence of primes $p$ such that satisfy the condition $$\gcd(p,S(p))>1$$ starts as $2,11,17,2161,\ldots$. Can you provide us any idea about if this sequence have finitely many terms? Many thanks.

Just to emphasize, since I'm asking two questions, only is required that you provide a cheap bound for A) and a suitable reasoning/heuristic for B), to get idea for these problems.

Computational evidence and documentation for Question B. We've the following script in Pari/GP showing the first few terms

for(n=1, 10000, if(gcd(n,sum(k=1,n,n%k))>1&&isprime(n)==1,print(n)))

that you can evaluate on the website Sage Cell Server choosing as Language GP. Here the string sum(k=1,n,n%k) is our sum of remainders $S(n)$ with n%k coding $n\text{ mod }k$ for each integer $1\leq k\leq n$.

The asymptotic of $|\{1\leq n\leq x|\gcd(n,S(n))=1\}|$, with $S(n)$ the sum of remainders, and get idea for other miscellany problem

Let $n\geq 1$ be an integer. In this post we denote the sum of remainders function as $$S(n)=\sum_{k=1}^n n \bmod k,$$ for example $S(1)=S(2)=0+0$ and $S(5)=0+1+2+1+0=4$. In the literature there are problems that were studied related to the condition $\gcd(n,f(n))=1$, for a given arithmetic function $f(n)$.

Question. 

A) Is it possible to provide roughly a cheap bound for the cardinality $$\#\{1\leq n\leq x|\gcd(n,S(n))=1\}$$ as $x$ grows to $\infty$? 

B) The sequence of primes $p$ that satisfy the condition $$\gcd(p,S(p))>1$$ starts as $2,11,17,2161,\ldots$. Can you provide us any idea about if this sequence has finitely many terms?

Just to emphasize, since I'm asking two questions, only is required that you provide a cheap bound for A) and a suitable reasoning/heuristic for B), to get idea for these problems.

Computational evidence and documentation for Question B. We've the following script in Pari/GP showing the first few terms

for(n=1, 10000, if(gcd(n,sum(k=1,n,n%k))>1&&isprime(n)==1,print(n)))

that you can evaluate on the website Sage Cell Server choosing as Language GP. Here the string sum(k=1,n,n%k) is our sum of remainders $S(n)$ with n%k coding $n \bmod k$ for each integer $1\leq k\leq n$.

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user142929
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A deduction about the asymptotic of $|\{1\leq n\leq x|\gcd(n,S(n))=1\}|$, with $S(n)$ the sum of remainders, and get idea for other miscellany problem

Let $n\geq 1$ be an integer, in this post we denote the sum of remainders function as $$S(n)=\sum_{k=1}^n n\text{ mod }k,$$ for example $S(1)=S(2)=0+0$ and $S(5)=0+1+2+1+0=4$. In the literature there are problems that were studied related to the condition $\gcd(n,f(n))=1$, for a given arithmetic function $f(n)$.

Question. A) Is it possible to provide roughly a cheap bound for the cardinality $$\#\{1\leq n\leq x|\gcd(n,S(n))=1\}$$ as $x$ grows to $\infty$? B) The sequence of primes $p$ such that satisfy the condition $$\gcd(p,S(p))>1$$ starts as $2,11,17,2161,\ldots$. Can you provide us any idea about if this sequence have finitely many terms? Many thanks.

Just to emphasize, since I'm asking two questions, only is required that you provide a cheap bound for A) and a suitable reasoning/heuristic for B), to get idea for these problems.

Computational evidence and documentation for Question B. We've the following script in Pari/GP showing the first few terms

for(n=1, 10000, if(gcd(n,sum(k=1,n,n%k))>1&&isprime(n)==1,print(n)))

that you can evaluate on the website Sage Cell Server choosing as Language GP. Here the string sum(k=1,n,n%k) is our sum of remainders $S(n)$ with n%k coding $n\text{ mod }k$ for each integer $1\leq k\leq n$.