# Questions tagged [divisors-multiples]

For questions on divisors and multiples, mainly but not exclusively of integers, and related and derived notions such as sums of divisors, perfect numbers and so on.

22 questions
350 views

### If $q^k n^2$ is an odd perfect number with Euler prime $q$, is $\sigma(q^k)/n + \sigma(n)/q^k$ bounded from above?

STATEMENT OF THE PROBLEM If $q^k n^2$ is an odd perfect number with Euler prime $q$, is $\sigma(q^k)/n + \sigma(n)/q^k$ bounded from above? MOTIVATION Let $\sigma=\sigma_{1}$ denote the classical ...
399 views

### On comparing two almost injective divisor maps

Edit 2018.08.08 This answer https://mathoverflow.net/a/307881 will be updated to give recent information about S, especially a forthcoming preprint. End Edit 2018.08.08 In an introductory post on ...
363 views

172 views

### A conjecture regarding odd perfect numbers

(Note: I asked this question in MSE this June 2018 but did not receive any responses there. I have therefore cross-posted it here, hoping that it gets answered.) Let $\sigma(z)$ denote the sum of ...
304 views

125 views

### Is there an integer $r \neq q$ (with $r>1$) such that $N = q^k n^2 = \frac{r(r+1)}{2}\cdot{d}$ is an odd perfect number with $d>1$?

Slowak showed in 1999 that every odd perfect number $N = q^k n^2$ can be written in the form $$N = \dfrac{{q^k}\sigma(q^k)}{2}\cdot{D}$$ where $D>1$. From this result, it follows that every odd ...
I need at least basic information about generating functions of the following class of arithmetic functions, grouped by levels $k$. Fix some $k$ and some family $\varepsilon_*=(\varepsilon_\sigma)_{\... 0answers 232 views ### On even almost perfect numbers other than powers of two (Note: This question is an improved version of and has been cross-posted from this MSE post.) Let$\sigma(x)$denote the sum of the divisors of$x$. If$\sigma(x) = 2x - 1$, then we call$x$an ... 0answers 104 views ### Consecutive integers divisible by consecutive small numbers Given$n$, what is the largest set of consecutive integers in$[n,2n]$can we have so that each integer is divisible by a distinct element from$[\log n,2\log n]$(no partiular order)? So apriori I am ... 0answers 90 views ### gcd of polynomial values Suppose that$f$and$g$are two coprime polynomials in$\mathbb Z[x]$. I'm interested in any sort of upper bound on$gcd(f(a),g(a))$, in terms of the integer$a$. Are there any results of this type?... 0answers 90 views ### Eigenvalues of a sequence of matrices involving the divisor function Let$A_{n,k},k=1,\ldots,n$be a sequence of$n\times n$upper triangular matrices where$A_{n,1}=I_n$and$A_{n,k},\quad 2\leq k\leq n$be a regularly shifted and scaled matrix, with$P_{n,k}$an$n\...
In a previous mathoverflow question here a construction of a primitive sequence $1<a_1<\cdots<a_k\leq n$ formed by including all the integers in $[1,n]$ with exactly $k$ prime divisors (...