Questions tagged [arithmetic-functions]
For questions on arithmetic functions, i.e. real or complex valued functions defined on the set of natural numbers.
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Is there an algorithm to solve number manipulation problems?
There are this kind of problems where, given a certain amount of the same numbers, it's needed to manipulate them with functions or operators in a way to get a certain result. It's allowed to glue ...
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Is this proof for the divisibility constraint $\sigma(q^k)/2 \mid n$ correct, where $q^k n^2$ is an odd perfect number with special prime $q$?
Let $N = q^k n^2$ be an odd perfect number given in Eulerian form, i.e. $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Define the GCDs
$$G = \gcd(\sigma(q^k),\...
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On the equations $D(q^k)D(n^2)=2s(q^k)s(n^2)$ and $\sigma(q^k)\sigma(n^2)=2 q^k n^2$, where $q^k n^2$ is an odd perfect number with special prime $q$
MOTIVATION
The topic of odd perfect numbers likely needs no introduction.
In what follows, we denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, the ...
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Arithmetic Progression/Sequence | Does it include terms prior to the initial term?
I’m trying to find the right term to describe points along a linear equation f(x) = mx + b, but where the domain is restricted to the set of integers. EDIT: Also the slope and y-intercept are integers....
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Is this disproof for the Descartes-Frenicle-Sorli Conjecture that $k=1$, if $p^k m^2$ is an odd perfect number, valid?
Let $N = p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$.
It is known that
$$D(p^k)D(m^2)=2s(p^k)s(m^2) \tag{0}$$
where $D(x)=2x-\...
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Long lists of values of arithmetic functions
I'm looking to do some numerical work on values of various arithmetic functions, such as $d(n)$ (the number of divisors of $n$), $\sigma(n)$ (the sum of divisors of $n$), and $\varphi(n)$ (Euler's phi ...
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Weird arithmetics with prime numbers...
Hello here is a problem with which I struggle quite a bit:
For any integer $n \geq 2$, with prime factor decomposition $n = p_1^{α_1}\times p_2^{α_2}\times ... p_k^{a_k}$, let $f(n) = α_1^{p_1}\times ...
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Is the set of divisor indicator functions at $x^2 - 1$: $B' = \{(d \mid x^2 - 1) : d \in \Bbb{N}\}$ a $\Bbb{Z}$-linearly independent set?
We know that the set $B = \{(d \mid x) = \begin{cases} 1, d \mid x \\ 0, \text{ otherwise} \end{cases}, \ d \in \Bbb{N}\}$ forms a $\Bbb{Z}$-module Schauder basis for the module $M =\{ \Bbb{N} \to \...
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Is there an analytical solution to the inequality $\frac{1 + k({p}^{(k+1)/2)})}{p^k} < \frac{p}{p - 1}$ if one were to bound $k$ in terms of $p$?
My question is as is in the title:
Is there an analytical solution to the inequality
$$\frac{1 + k({p}^{(k+1)/2)})}{p^k} < \frac{p}{p - 1}$$ if one were to bound $k$ in terms of $p$?
Here, $p \...
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Why is it that, if there are no odd perfect numbers, then there are no other $3$-perfect numbers, apart from the six known, as of the year $1643$?
Let $\sigma(x)=\sigma_1(x)$ denote the classical sum of divisors of the positive integer $x$.
A number $N$ is said to be $k$-perfect if $\sigma(N)=kN$ where $k$ is a positive integer.
The number $1$ ...
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What is the Dirichlet serie of The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$?
The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$ and defined by
$$
A(n)=\sum \limits_{p^{\alpha}\parallel n}\alpha p
$$
is this serie calculated ...
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Is this logical deduction regarding some modular restrictions on odd perfect numbers valid?
Let $p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)...
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Characterization of Möbius-monotonicity
We say that an arithmetic function $f:\mathbb{Z}^+\to\mathbb{C}$ is Möbius-monotone if $\forall n\geq1:(\mu*f)(n)\geq0$ (where $*$ denotes de Dirichlet convolution), i.e. if there exists a non-...
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On the prime factorization of $n$ and the quantity $J = \frac{n}{\gcd(n,\sigma(q^k)/2)}$, where $q^k n^2$ is an odd perfect number
Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\...
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On a consequence of $G \mid I \iff \gcd(G, I) = G$ (Re: Odd Perfect Numbers and GCDs)
Let $N = q^k n^2$ be an odd perfect number given in the so-called Eulerian form, where $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of ...
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Does $\sigma(n^2)/q \mid q^k n^2$ imply $\sigma(n^2)/q \mid n^2$, if $q^k n^2$ is an odd perfect number with special prime $q$? - Part III
(Preamble: This question is an offshoot of the following inquiry in MSE.)
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$...
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Show the function for which the Dirichlet generating series is $\zeta(2s)$ using only $\tau,\varphi,\sigma\text{ and }\mu$ or some explicit formula.
I'm trying to find the function with Dirichlet generating series $\zeta(2s)$, I know that this relates somehow to the Liouville function but I am trying to express it in terms of only the standard ...
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Generating function for the "two variable totient"?
Recall that Euler's totient function $\varphi(n)$ is defined as the number of natural numbers $<n$ which are coprime to $n$.
The Dirichlet series generating function for $\varphi(n)$ is
$$
\sum_{n\...
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What's so special about the number 24 in the definition of the Ramanujan tau function?
I'm learning about the Ramanujan tau function $\tau \colon \mathbb{N} \to \mathbb{Z}$ defined by $$\tau(1)q + \tau(2)q^2 + \tau(3)q^3 + \ldots = q \prod_{n = 1}^{\infty} \left(1 - q^n\right)^{24}.$$
...
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Product of all reduced residues in relation with the function $\mu$: $\prod_{1\leq a\leq n,(a,n)=1}a=n^{\varphi(n)}\prod_{d|n}(d!/d^d)^{\mu(n/d)}$
We know that for any arithmetic function $f$ the Mobius inversion formula gives its inversion. Hence
$$
F(n)=\prod_{d|n}f(d)\implies f(n)=\prod_{d|n}
F(n/d)^{\mu(d)}.$$
The above statement can be ...
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Can this inequality involving the deficiency and sum of aliquot divisors be improved? - Part II
This MSE question (from April 2020) asked whether the inequality
$$\frac{D(n^2)}{s(n^2)} < \frac{D(n)}{s(n)}$$
could be improved, where $D(x)=2x-\sigma(x)$ is the deficiency of the positive integer ...
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Which arithmetic functions satisfy $\sum_{d \mid n} f(\frac{n}{d}) k^d \equiv 0 \pmod{n}$ for all positive integers $n,k$?
There are $$\frac{1}{n} \sum_{d \mid n} \varphi\left(\frac{n}{d}\right) \cdot k^d$$ different $k$-ary necklaces of length $n$ as a result of Pólya's enumeration theorem, where $\varphi$ is Euler's ...
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On the equation $s(n^2) = \left(\frac{q-1}{2}\right)\cdot{D(n^2)}$, if $q^k n^2$ is an odd perfect number with special prime $q$
Let $N$ be an odd perfect number given in the so-called Eulerian form
$$N = q^k n^2$$
where $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
In what follows, let ...
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On the second moment of prime divisor function
Let $\omega(n)$ denote the number of distinct prime divisors of $n$. I learned from Cojocaru & Murty that
$$
\sum_{n\le x}\omega(n)^2=x(\log\log x)^2+O(x\log\log x).\tag1
$$
I wonder whether it is ...
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If $2^r b^2$ is an even almost perfect number that is NOT a power of two, does it follow that $r=1$?
(The following are taken from this preprint by Antalan and Dris.)
Antalan and Tagle showed that an even almost perfect number $n \neq 2^t$ must necessarily have the form $2^r b^2$ where $r \geq 1$, $\...
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Does $\sigma(n^2)/q \mid q^k n^2$ imply $\sigma(n^2)/q \mid n^2$, if $q^k n^2$ is an odd perfect number with special prime $q$? - Part II
(Preamble: This inquiry is an offshoot of this MSE question.)
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$ as $I(x)=\...
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Has this strange real-valued binary function been researched?
Let $X \neq \emptyset$ be an arbitrary set. Let
$$ d : X \times X \to \mathbb{R} $$
be a real-valued function with the following properties:
(1) $\forall x, \in X: d(x,x) = 0,$
(2) $\forall x,y \in X:...
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Are these valid proofs for the equation $\gcd(n,\sigma(n^2))=\gcd(n^2,\sigma(n^2))$, if $q^k n^2$ is an odd perfect number with special prime $q$?
Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$.
It is known that
$$i(q)=\gcd(n^2,\sigma(n^2))=\frac{\sigma(n^2)}{q^k}=\frac{n^2}{\sigma(q^k)/...
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On a curious PDE involving $I(q^k)+\frac{2}{I(q^k)}$
Let $I(x)=\sigma(x)/x$ be the abundancy index of the positive integer $x$, where $\sigma(x)=\sigma_1(x)$ is the classical sum of divisors of $x$.
Consider
$$f(q,k)=I(q^k)+\frac{2}{I(q^k)}$$
where $5 \...
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Are the (Bezout) coefficients for $\gcd(n^2, \sigma(n^2)) = q\sigma(n^2) - 2(q - 1)n^2$ (where $q^k n^2$ is an odd perfect number) unique?
In what follows, denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, the deficiency of $x$ by $D(x)=2x-\sigma(x)$, and the aliquot sum of $x$ by $s(x)=\sigma(x)...
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Why counting the number of 1 digits that appear in all integers in 0-9, 0-99, 0-999, 0-9999 follow an arithemtic-geometric sequence?
I noticed that
0-9 = has only 1 '1'
0-99 = has 20 '1's [1,10,11,12,13,14,15,16,17,18,19,21,31,41,51,61,71,81,91]
0-999 = 300
0-9999 = 4000
It follows the formula of
n = number of digits in the ...
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If $p$ and $2p + 1$ are odd primes and $n = 4p$, prove that $φ(n + 2) = φ(n) + 2$. [closed]
Prove: if $p$ and $2p + 1$ are odd primes and $n = 4p$, show that $φ(n + 2) = φ(n) + 2$.
I'm stuck on this simple question about Euler's theorem.
Any help would be welcome.
Thanks in advance!
user1079281
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Is the argument used in this proof that $k=1$ logically sound, where $q^k n^2$ is an odd perfect number with special prime $q$?
The topic of odd perfect numbers likely needs no introduction.
Euler proved that a hypothetical odd perfect number $N$, if one exists, must have the so-called Eulerian form $N=q^k n^2$, where $q$ is ...
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$\varphi(x,n)=\sum_{d|n}\mu(d)\left\lfloor\frac xd\right\rfloor$ and $\sum_{d|n}\varphi\left(\frac xd,\frac nd\right)=\lfloor x\rfloor$
If $x$ is real, $x\ge 1$, let $\varphi(x,n)$ denote the number of positive integers less than or equal to $x$ that are relatively prime to $n$. Prove that
$$\varphi(x,n)=\sum_{d|n}\mu(d)\left\lfloor\...
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Product form of Mobius Inversion formula: $g(n)=\prod_{d|n}f(d)^{a(n/d)}\iff f(n)=\prod_{d|n} g(d)^{b(n/d)}$
Product form of the Möbius inversion formula: If $f(n)>0$ for all $n$ and if $a(n)$ is real, $a(1)\neq 0$, prove that
$$g(n)=\prod_{d|n}f(d)^{a(n/d)}\iff f(n)=\prod_{d|n} g(d)^{b(n/d)}$$
where $b=a^...
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Prove that $\sum_{d|n}\mu(d)\log^m d=0$
Prove that
$$\sum_{d|n}\mu(d)\log^m d=0$$
if $m\ge 1$ and $n$ has more than $m$ distinct prime factors.
I tried using Induction and kind of succeeded in the sense that if we write down the case $m=2$ ...
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If $q^k n^2$ is an odd perfect number with special prime $q$, is it possible to express $\sigma(n^2)/q^k$ as a function of only $q$ and $k$?
Let $q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)...
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Relation between Mertens function and Chebyshev function
In Wikipedia article about Mertens function we can read: "A curious relation given by Mertens himself involving the second Chebyshev function is
$$\displaystyle \psi (x)=\sum _{n=1}^{\infty } M\...
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Slight variation of Mertens' third theorem. Do we have an estimate of $\sum_{d \mid n\#}(-1)^{\omega(d)}\dfrac{\sigma_0(d)}{d}$?
Define $f(n)$ to be:
$$
\sum_{d \mid n\#}(-1)^{\omega(d)}\dfrac{\sigma_0(d)}{d}
$$
But $\sigma_0(d) = 2^{\omega(d)}$ for any $d \mid n\#$ a primorial, so:
$$
f(n) = \prod_{p \text{ prime} \\ p \leq n} ...
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Does the following lower bound improve on $I(q^k)+I(n^2) > \frac{57}{20}$, where $q^k n^2$ is an odd perfect number?
Preamble: This question is an offshoot of this earlier post. (This inquiry has likewise been cross-posted to MO last June $10, 2022$.)
Let $N = q^k n^2$ be an odd perfect number with special prime $q$...
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Follow-up to MSE question 3738458
This is a follow-up inquiry to this MSE question.
Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$.
A number $M$ is said to be perfect if $\sigma(M)=2M$. For example, $6$ and $...
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What are the fixed points of the arithmetic derivative over the non-negative integers?
I just watched When the derivative of a number is not zero -- The arithmetic derivative by Michael Penn.
I like to explore things visually and computationally, so I found this recursive implementation ...
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Full derivation inside of twin prime statement in terms of multiplicative arithmetic functions. How can the last formula be rearranged?
Let $(\cdot\mid\cdot) : \Bbb{N}\times\Bbb{N} \to \Bbb{Z}_2$ be the divisibility function which takes on the value $(x|y) = 1$ whenever $x$ divides $y$ and the value $(x|y) = 0$ whenever it does not ...
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Characterization of primes of the form $n^n+1$ by using number-theoretic functions
It is known that there is a unsolved problem related to primes of the form $n^n+1$ as is explained in page 160 of [1] (see also page 156, and the OEIS page related to this integer sequence A121270). ...
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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 ...
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Using Mobius Inversion Formula
We are given the following:
$f(n)=\prod_{d|n}g(d)$
and asked to show:
$g(n)=\prod_{d|n} f(d)^{\mu(\frac{n}{d})}$
The hint given says to use logarithms
Here's what I tried doing:
$log(f(n))=\prod_{d|n}...
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Does $I = \gcd(n,\sigma(n^2)) = (\frac{n}{\sigma(q^k)/2})\cdot\gcd(\sigma(q^k)/2,n)$ imply that $\sigma(q^k)/2 \mid n$ holds?
Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Define the GCDs:
$$G = \gcd\bigg(\sigma(q^k),\sigma(n^2)\bigg)$$
$$H = \...
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Different ways to average an arithmetic function
Consider the following ways to average a multiplicative arithmetic function $f$ over $\mathbb{N}$:
Arithmetic: $\displaystyle\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N f(n)$
Factorized: $\displaystyle\...
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The number of solutions $(x,y)$ of the congruence $n \equiv X^2-XY+Y^2$ (mod $p^\alpha$).
Is there general formula of the solutions of the congruence?
\begin{equation}
n\equiv X^2-XY+Y^2 \pmod r,
\end{equation}
where $n\in\Bbb Z$ and $r\in\Bbb N$.
If we define an arithmetic function (two ...
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Investigating $\sum \prod_{p\mid n}(1-\frac{1}{p^2})$ as $x\to\infty$
I'm investigating the behavior of the following function as $x\to \infty$:
$$f(x):=\sum_{1\le n\le x}\frac{J_2(n)}{n^2}$$
where $J_k(n)$ is the Jordan's totient function
$$J_k(n):=n^k\prod_{p\mid n}(1-...
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