# Questions tagged [divisor-sum]

For questions on the divisor sum function and its generalizations.

772 questions
Filter by
Sorted by
Tagged with
71 views

### What conditions on $X$ will guarantee that $\gcd(\text{square part of } X,\text{squarefree part of } X)=1$, if $X$ is neither a square nor squarefree?

The following query is an offshoot of this answer to a closely related post. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. The topic of odd perfect ...
• 10.2k
22 views

### Exponential sums with the divisor function

Can anyone explain the first $\ll$ on line 8 on page 188 of "Jutila: On exponential sums involving the divisor function" for me? (https://eudml.org/doc/152693) Specifically, I think he is ...
• 1,323
1 vote
46 views

### On Carmichael function and aliquot parts of odd perfect numbers

We denote as $N$ an odd perfect number, and $d\mid N$ one of its divisors. We denote the Carmichael function as $\lambda(x)$, Wikipedia has the article Carmichael function dedicated to this number ...
44 views

• 6,772
1 vote
108 views

### Does there exist a nontrivial prime power $q^k$ such that $\sigma(n^2)/n = q^k$ for some $n$?

Let $\sigma(x)=\sigma_1(x)$ be the classical sum of divisors of the positive integer $x$. My question in the present post is closely related to this one in MO: QUESTION Does there exist a nontrivial ...
• 10.2k
19 views

### R.T.P. σ(p^n)=(p^(n+1)-1)/(p-1). (where σ denotes the divisor function) For a prime p. [closed]

I need to prove that for a prime p, σ(p^n) is (p^(n+1)-1)/(p-1). E.g. σ(3^3) is (3^(3+1)-1)/(3-1). =3^(4)-1/2 =80/2 =40 Therefore σ(3^3)=40. Let me know for any suggestions,proofs or references.
116 views

### On odd perfect numbers and a GCD - Part VII

(Pardon me for being somewhat stubborn, but this question will be the last for this week. This post is an offshoot of this one.) Let $N = q^k n^2$ be an odd perfect number be an odd perfect number ...
• 10.2k
38 views

• 10.2k
77 views

### On improving the upper bound $I(m^2) \leq \frac{2p}{p+1}$, if $p^k m^2$ is an odd perfect number with special prime $p$

(Preamble: This question is an offshoot of this earlier MSE post.) Let $p^k m^2$ be an odd perfect number (OPN) with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$. ...
• 10.2k