# 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.

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### A definition related to pseudoprimes and the Dedekind psi function

In this post we consider that $\psi(k)$ denotes the Dedekind psi function. Wikipedia has an artcle dedicated to this arithmetic function Dedekind psi function defined for a positive integers $m>1$ ...
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### Around the equation $\sigma\left(\square\right)=\text{prime}$: counterexamples or a proof for some of these conjectures

For integers $A,B\geq 1$ we define the difference $\sigma(A)\sigma(B)-\sigma(AB)$, denoting it as $[A,B]$, where $\sigma(n)=\sum_{1\leq d\mid n}d$ denotes the sum of divisors function. It is possible ...
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### Have any proposals been advanced for the analytic continuation of the divisor function?

While I was working on the evaluation of a certain series, the following limit came up: \begin{align} \lim_{n \to 1} \frac{d(n)-1}{n(n-1)} &= \lim_{n \to 1} \frac{d'(n)}{2n-1} \\ &= d'(1) .\...
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In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$, satisifies the following recurrence relation ($n>1$): $$n^2(n-1) = \frac{6}{\sigma(n)} \sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\... 5 votes 0 answers 438 views ### Does the equation \sigma(\sigma(x^2))=2x\sigma(x) have any odd solutions? This question was posted in MSE in early August 2020. It did garner several upvotes, but did not receive any responses. I have therefore cross-posted it here, hoping that it gets answered. Let \... 2 votes 1 answer 151 views ### Sum of divisors and unitary divisors as the eigenvalue and the spectral norm of some addition matrix? Let n be a natural number and D_n be the set of divisors. We can make this set to a ring by observing that each divisor d has$$0 \le v_p(d) \le v_p(n) Hence we can add two divisors $d,e$ by ... (This post is an offshoot of this MSE question.) Let $\sigma(x)$ denote the sum of divisors of $x$. (https://oeis.org/A000203) QUESTION Is the asymptotic density of positive integers $n$ satisfying $... 6 votes 1 answer 339 views ### If$n = 18k+5$is composite, there are at least 9 divisors of$\phi(n)$which do not divide$n-1$If$n$is a composite of the form$18k+5$, there at least 9 divisors of$\phi(n)$which do not divide$n-1$. Is this true in general or if not, what is the smallest counter example? The conjecture has ... 1 vote 0 answers 98 views ### If$n$is a multiperfect number, then necessarily does one of its prime factors$p$satisfy$p \parallel n$? My question is as in the title: If$n$is a multiperfect number, then necessarily does one of its prime factors$p$satisfy$p \parallel n$? I quote from an answer by Varun Vejalla to a closely ... 6 votes 0 answers 261 views ### Is there a positive odd$n$such that$\sigma(\sigma(n)) = \sigma(\sigma(n)-n)+\sigma(n)$? Let$\sigma(n)$denote the sum of the divisors of$n$. (https://oeis.org/A000203) It is relatively easy to find numbers$n$such that$f(g(n)) = g(f(n))$where$f(n) = \sigma(n)$and$g(n) = \sigma(n) ...
This question is related to sequence of numbers $t$ such that $F_{6t}$ is a nontotient where $F_n$ represents the sequence of Fibonacci numbers for $n\geq 0$. The online encyclopedia Wikipedia has the ...