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.

82 questions
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Numbers with all N-digit prefixes divisible by N

In base 10, the number 3816547290 contains every digit exactly once. When I take the first N digits, that substring is divisible by N. For example, 381 is divisible by 3, 38165 is divisible by 5, etc. ...
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Is there a formula that can predict the primes in the sequence of ratios of consecutive superior highly composite numbers? : $2, 3, 2, 5, 2, 3, 7,…$

This is the sequence of prime numbers which are the elementary building blocks for the superior highly composite numbers: $2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 2, 23, ...$ The $n^{th}$ ...
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Does iterating a certain function related to the sums of divisors eventually always result in a prime value?

Let define the following function for integers (from 2): $f(x)=\sigma(x)-1$, where $\sigma$ is the sum of the divisors of $x$. For example $f(6)=6+3+2=11$, $f(5)=5$. Note that $x$ is a fixed point for ...
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Arbitrarily large $n$ divides $F_n$

Is it true that there exists $n \in \mathbb{N}$ with arbitrarily many prime factors such that $n$ divides $F_n$, where $F_n$ represents the n-th Fibonacci number?
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runs of consecutive non squarefree integers

This question gained no attention at Math SE. Call a sequence of $k$ consecutive naturals squary if each one of them is divided by a square > 1. The Chinese Remainder theorem trivially guarantees us ...
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Who is attributed with the conjecture that every multiply-perfect number greater than $1$ is even?

I know that Descartes is considered to be the first to ask whether or not odd perfect numbers exist ($n$ such that $\sigma(n)=2n$, where $\sigma(n)$ is the sum of divisors of $n$), and he also ...
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Sum of divisor function over arithmetic progression

I am trying to find an estimate for the following sum: $$\sum_{\substack{n \leq x \\ n \equiv k (m)}} d(n),$$ where $d(n)$ is number of divisors of $n$. I found estimates for the case when $k$ and ...
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Determine whether if $n$ is a primitive pseudoperfect (semiperfect) number, then $\sigma(n)<2^{\sigma_0(n)}$

Determine whether if $n$ is a primitive pseudoperfect (semiperfect) number, then $\sigma(n)<2^{\sigma_0(n)}$. $\sigma_k(n)$ is the division function and $\sigma(n)=\sigma_1(n)$. A number is ...
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Are all known $k$-multiperfect numbers (for $k > 2$) not squarefree?

I asked the following question in MSE four ($4$) days ago, but so far nobody has posted an answer. The gist of the question is as follows: Are all known $k$-multiperfect numbers (for $k > 2$...
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What are the divisors of $2n^2 - \sigma_{1}(n^2)$ for composite $n$?

What are the divisors of $2n^2 - \sigma_{1}(n^2)$ for composite $n$? Here, $\sigma_{1}$ is the classical sum-of-divisors function. For example, $\sigma_{1}(3^2) = 1 + 3 + {3^2} = 13$. (The function ...
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Strange (or stupid) arithmetic derivation

Let us consider the following operation on positive integers: $$n=\prod_{i=1}^{k}p_i^{\alpha_i} \qquad f(n):= \prod_{i=1}^{k}\alpha_ip_i^{\alpha_i-1}$$ (Is it true that if we apply this operation to ...
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Divisibility properties of a stream of numbers

Let $S$ be a set of $k$ distinct natural numbers, each from the interval $[2,n]$, with least common multiple $\mathop{lcm}$. What fraction $\rho$ of the numbers $2,3,4,\ldots,\mathop{lcm}$ are ...
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Maximal order of Hooley's Delta function?

There is a large literature on Hooley's $$\Delta(n)=\max_u\sum_{d|n,\ e^u\le d< e^{u+1}}1$$ giving its normal and average order. What is known of its maximal order? Clearly $\Delta(n)\le d(n)$ ...
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Finding primes using Euler's sum of divisors recurrence relation

Euler came up with following recurrence relation for the sum of divisors (refer to http://arxiv.org/abs/math/0411587) $$\sigma(n) = \sigma(n−1) + \sigma(n−2) − \sigma(n−5) − \sigma(n−7) \dots$$ Since ...
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Why do primes dislike dividing the sum of all the preceding primes?

I was investigating primes with the property that the sum of the first $n$ primes is divisible by $p_n$. It turns out that these primes are extremely extremely rare. For primes less than $10^9$, I ...
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Ratio of consecutive divisors and average

Let $2\leq d_1 < d_2,...,d_l < n$ be all the proper nontrivial divisors of $n$. I like to understand how much these divisors deviates from each other. Here are two questions in this regard: (1) ...
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Is it necessary that gcd > 1 of an infinite set? [closed]

Consider an infinite set $S$, of positive integers. If all the finite subsets of $S$ have GCD $>$ $1$, is it necessary that the GCD of $S$ is greater than $1$ as well?
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Getting a bound on the coefficients of the factor polynomial

Suppose $f(x):=a_0+a_1x+\cdots+a_nx^n$ is a polynomial in $\mathbb{Z}[x]$ and $|a_i|\leq M$ for each $i=0,\ldots ,n.$ Now suppose $g(x)$ is a factor of $f(x)$ in $\mathbb{Z}[x]$, then is it possible ...
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On Robin's criterion for RH [closed]

$$\sigma(n) < e^\gamma n \log \log n$$ In 1984 Guy Robin proved that the inequality is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin 1984)....
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What literature is known about MacMahon's generalized sum-of-divisors function?

MacMahon in the paper Divisors of Numbers and their Continuations in the Theory of Partitions defines several generalized notions of the sum-of-divisors function; for example, if we write $a_{n,k}$ ...
Hello all, if $a_1,a_2, \ldots a_t$ are $t$ integers $\geq 2$, the set $G(a_1,a_2, \ldots a_t)=\lbrace N \geq 1 |$ In any sequence of $N$ consecutive integers there is at least one not divisible by ...