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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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  • $\begingroup$ Thanks you very much @Lspice $\endgroup$ – user142929 Jan 30 at 8:38
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This observation might be already in the literature, but I have not seen it written down before.

For your second question: Since we do not suspect any "obvious" arithmetical relation between $p$ and $S(p)$, let us guess that $S(p)$ takes a random residue class $\mod p$; that is, it has a probability of $1/p$ of being a multiple of $p$. Then we would expect about $\sim \log \log x$ primes $p$ up to $x$ for which $p$ divides $S(p)$: so, infinitely many of them, but very sparse. I cannot find any beyond $4441$.

For lack of a better guess, let us also stipulate that $S(n)$ is a random integer (of its size, so $\sim cn^2$). In particular, its residue class $\mod n$ should also be random. But there are asimptotically $6/\pi^2$ such classes that are coprime to $n$--at least, if we also sum over $n \leq x$. Hence, $S(n)$ has a probability of $6/\pi^2$ of being coprime to $n$.

I am not saying that these heuristics are in any way well informed; but they agree well with numerical computation. Of course, this is not special to $S(n)$: any arithmetical function which does not have any particular relationship with $n$ itself will behave in the same way. The cases that are usually studied in the literature (say, $S(n)=\textrm{sum of $S$-units}$) are more interesting exactly because of nontrivial arithmetical relationships between $n$ and $S(n)$, which are enough to change the outcome.

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  • $\begingroup$ Many thanks for your answer and attention, I'm going to study it. $\endgroup$ – user142929 Jan 31 at 21:03

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