For questions on arithmetic functions, a real or complex valued function $f(n)$ defined on the set of natural numbers.

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1answer
31 views

Sums involving square of Moebius function

I try to estimate the following sum: $$ \sum_{n \leq x}\mu(n)^2 f(n) $$ where $\mu(n)$ is a Moebius function and $f(n)$ is some multiplicative arithmetic function. If I understand it correctly it is ...
2
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1answer
28 views

A Möbius Identity

I need to prove the identity $$\sum_{k = 1}^n \mu(k)\left[ \frac{n}{k} \right] = 1$$ where $n$ is a natural number, and $[n]$ denotes the floor function. The proof also should not use the Möbius ...
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0answers
29 views

Number of excellent pairs is equal to $\sigma(n)$

Let $\nu$ be an irrational positive number, and let $m$ be a positive integer. A pair of $(a,b)$ of positive integers is called good if $$a \left \lceil b\nu \right \rceil - b \left \lfloor a \nu ...
4
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1answer
65 views

Which natural number predicates can be defined in Robinson arithmetic?

I'm especially wondering about the order relation, subtraction, division and exponentiation here: $x \leq y \quad \Leftrightarrow \quad \exists u\ y=x+u$ $z= x-y \quad \Leftrightarrow \quad ...
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1answer
42 views

Find the term of the G.P.

Find the term of the G.P. $1, 1.2, 1.44, 1.728,... $ which is just greater than $100$. Please somebody explain me the question? All i know is $a=1$ and $r=1.2$ Help :( should it be $T_{n} > ...
4
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2answers
62 views

prove that $\phi(xy) =\phi(x)\phi(y)$ for any $x$ and $y$ with $(x, y) = 1$. [duplicate]

Prove that $\phi(xy) = \phi(x) \phi(y)$ for any $x$ and $y$ with $(x, y) = 1$. I understand the concept, and have done several examples proofing this but cannot put it in "proof form" because unless ...
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1answer
61 views

Finding formulas for sums

I know that $\sum_{d \mid n} \mu(d) = 0$ whenever $n >1$, and I know that $\sum_{d \mid n} \phi(d) = n$. How can I use this in order to give a formula for $\sum_{d \mid n} \mu(d)\phi(d)$?
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1answer
81 views

Without using group theory, How to Prove $n|\phi(a^n-1)$, where $\phi$ is Euler's Totient function. [closed]

Let $\phi$ be Euler's Totient funcion, how to prove this property? If possible can we have an elementary proof without leveraging the group theory? $$n|\phi(a^n-1), \forall n,a>1, \gcd(a,n)=1$$ ...
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2answers
20 views

Find the numbers; Arithmetic Progression.

The sum of four consecutive numbers in an A.P is $28$. The product of the second and third numbers exceeds that of the first and last by $18$. Find the numbers. I thought of this: $$S_{4} = ...
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0answers
26 views

An inequality with $\omega(n)$ [duplicate]

Prove: For any positive integer $k, N$, $$\left(\frac{1}{N}\sum\limits_{n=1}^{N}\left(\omega (n)\right)^k\right)^{\frac{1}{k}}\leq k+\sum\limits_{q\leq N}\frac{1}{q}$$ Where $\sum\limits_{q\leq ...
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0answers
19 views

An inequality involving Möbius function [duplicate]

For any positive integer $n$ show the inequality holds : $$\left|\sum_{i=1}^{n}\frac{\mu(i)}{i}\right|\le 1$$ I tried induction. when $\mu(n+1)=0$ it is trivial. But what if $\mu(n+1)\ne 0$? I am ...
3
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1answer
68 views

A sum regarding prime factorization

Prime factorization of $n$ is $n = p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$ Let $f(n) = \left((p_1^{a_1}+1)(p_2^{a_2}+1)(p_3^{a_3}+1)\cdots(p_k^{a_k}+1)\right)$ I want to find the value of ...
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0answers
53 views

Summation of no. of divisors

Let $d(n)$ = no. of divisors of $n$ and $(d(n))^2$ = square of no. of divisors of n. Let $$S(N) = \sum_{n=1}^{N} d(n)$$ and $$S_2(N) = \sum_{n=1}^{N}(d(n))^2$$ $$S(N) = ...
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2answers
22 views

Mean of given Arithmetic function

Find the mean of $a, a+d, a+2d, a+3d,\dots,a+nd$ I have no idea what to do in this question but i have tried the following: $$mean\ \bar{x}= \frac{(a)+(a+d)+(a+2d)+(a+3d)+\cdots+(a+nd)}{n+1} $$ ...
4
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1answer
95 views

Euler phi function, number theory

I am trying to find the value of $$\sum_{n=1}^{N}\sum_{d|n}d*\phi(d)$$ Is there a method to evaluate this for large N? $\phi(d)$ is the Euler phi function.
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1answer
33 views

Arithmetic progressions that generate an infinite number of powers of 2?

For an arithmetic progression of the form $a_i = ki, i \in \mathbb{N}$, the question is trivial - if $k$ is a power of 2, then the progression will generate an infinite number of powers of 2, and no ...
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4answers
87 views

What's the meaning of this $(m,n) = 1$

I'm reading this pdf http://rutherglen.science.mq.edu.au/wchen/lndpnfolder/dpn01.pdf I understand some of the expression used in this but I don't understand the part $(m,n) = 1$ Is this a cartesian ...
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1answer
55 views

Farey Sequence Vector Orthogonality Relation Question

Take the Farey sequence $\mathcal{F}_n$ for $n=39$ with values $a_m\in \mathcal{F}_n$ and put them into a vector $$ \vec v_k=\biggr(\exp(2\pi i k a_m)\biggr)_m $$ Since Merten's function for $n=39$ ...
4
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3answers
101 views

Question about $T_n = n + (n-1)(n-2)(n-3)(n-4)$

The formula $T_n = n + (n-1)(n-2)(n-3)(n-4)$ will produce an arithmetic sequence for $n < 5$ but not for $n \ge 5$. Explain why. I think it is because if n is less than five the term with ...
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2answers
41 views

Finding root for the segment - found the formula but it doesn't work for some values - wrong formula?

I have the segment, defined as $(x_1, y_1)$, $(x_2, y_2)$. I know that $y_1\ge 0$ and $y_2 < 0$. I want to compute the root point for that segment. I decided to do it that way: ...
2
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1answer
48 views

Arithmetic function

An arithmetic function is defined as follows:$f(1)=1$, $f(2k)=k$ and $f(2k+1)=f(k)+f(k+1)$. When (for which $k$) is $f(k)$ even? While it is obvious that $f(4n-1)=f(4n)=2n$, therefore $f(k)$ is even ...
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1answer
69 views

What are arguments to $\frac 00 = Undefined$?

Now, I understand that dividing by zero in any case is undefined. However, in math, there are always exceptions. I'm just really curious...what are the different cases for different answers? For most ...
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0answers
117 views

Elementary references on Robinson Arithmetic, Prim. Recursive fns etc.

I'm in the middle of revising my freely available and much-downloaded introductory notes Gödel Without (Too Many) Tears. (They are a sort of cut down version of part of my Gödel book, and I'm ...
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1answer
16 views

how to check arithmetic progression of triangle

I solved the problem with the following text: In a rectangle the sides and the diagonal are an arithmetic progression. Calculate the circumference of the rectangle where the longer side is 44 cm ...
3
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1answer
24 views

Two sets with the same geometric and arithmetic means

There are two sets A and B with equal geometric mean and arithmetic mean. Each element of both sets is odd integer greater than 1. A = B ? Order of elements isn't important.
3
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1answer
79 views

does the surd $\sqrt[p-1]{p}$ for prime $p$ occur in any context?

for any prime $p \in \mathbb{N}$ use the corresponding symbol $q$ to denote the quantity $$q=p^{\frac1{p-1}}$$ and for $n \in \mathbb{N}$ define: $$ Q_n= \prod_{p \le n} q $$ empty products evaluate ...
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2answers
48 views

how to compute the “Hamming total” of a symmetric group?

The Hamming distance between two binary numbers has an obvious analogue for numbers encoded using a base $\gt 2$. A different analogue is the following. Let $\sigma \in S_n$ be a permutation which ...
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1answer
47 views

Evaluate $\sum_{d\mid N}\Lambda(d)$

For a positive integer $n$, define $$\Lambda(n) = \left\{ \begin{array} {ll} \log p & \mbox{if $n = p^r$, $p$ a prime and $r \in \mathbb{N},$ }\\ 0 & \mbox{otherwise.} ...
3
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1answer
71 views

Properties of Arithmetic Functions

I was recently working on arithmetic functions and using Perron's formula to obtain asymptotic estimates. One observation I made was that the Dirichlet series often can be written in terms of the ...
2
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1answer
73 views

A closed form for $\sum_{i\cdot j^k=n}(-1)^i$?

$$\alpha_k(n) \stackrel{\text{def.}}{=} \sum_{i\cdot j^k=n}(-1)^i.$$ Does a closed form exist for $\alpha_k(n)$? For low values of $k$: $$\alpha_0(n)=(-1)^n$$ $$\alpha_1(n)=\begin{cases} ...
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5answers
163 views

For every positive integer n greater than $2$, $\phi(n)$ is an even integer.

Theorem: For every positive integer n greater than $2$, then $\phi(n)$ is an even integer. I know this theorem and the same is used much, but I was curious how it would be to demonstrate it, show ...
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2answers
105 views

If $(m,n)=1$ then $m^{\phi(n)}+n^{\phi(m)}\equiv0\pmod {mn}$

Show that if $(m,n)=1$ then $m^{\phi(n)}+n^{\phi(m)}\equiv0\pmod {mn}$ I tried, $(m,n)=1$ then, Euler theorem we have $m^{\phi (n)}\equiv1\pmod n$ and $n^{\phi (m)}\equiv1\pmod m$. But I could ...
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2answers
174 views

Ramanujan 1918 paper

Does anybody know where I can find Ramanujan's paper from 1918 titled "On Certain Arithmetical Functions." It is referenced in wikipedia, under the Ramanujan Summation section, but I cannot find a ...
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5answers
131 views

Solve for $m\in\mathbb{N}$ the equation $\phi (m)=12$

Solve for $m\in\mathbb{N}$ the equation $\phi (m)=12$ I found (by trial) $m=\{13,21,26,28,36\}$, but do not know if misinterpreted the problem, but actually I suppose I have to find an equation ...
3
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3answers
79 views

Prove that for all $n\in\mathbb{N}$, $\frac{s(n)}{d(n)}\geq \sqrt n$

Prove that for all $n\in\mathbb{N}$ $$\frac{s(n)}{d(n)}\geq \sqrt n$$ where $s(n) = \sum_{d|n} d$ and $d(n) = \sum_{d|n} 1$. Being honest, study some time arithmetic functions, and can not ...
2
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4answers
114 views

Finding the GCD of $50!$ and $2^{50}$

I've been trying to figure out how $n!$ and $x^n$ are related (where x is an integer) for most of the morning - I know it must be the key to unlocking this problem. Up to this point I've only used ...
1
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1answer
36 views

Demonstration in arithmetic function

I need help to know (in detail) how to prove that the product of two multiplicative arithmetic functions is a multiplicative arithmetic function. $$$$$f(n)$ and $g(n)$ are functions multiplicative, ...
2
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1answer
46 views

Finding $m∈\mathbb N$ such that $d^n(m)$ is not a perfect square for any $n\geq1$

Let , for $k ∈\mathbb N$ , $d(k)$ denote the number of positive divisors of $k$ ; define $d^n (k)$ recursively as $d^1(k)=d(k)$ , for $n\geq1$ , $d^{n+1}(k)=d(d^n(k))$ , how do we find those ...
3
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1answer
55 views

Is there a friendly pair $\{a, b\}$ where both $a$ and $b$ are odd?

Let $\sigma(x)$ denote the sum of the divisors of a positive integer $x$. $\{y, z\}$ is said to be a friendly pair if $$I(y) = I(z),$$ where $I(x) = \sigma(x)/x$ is the abundancy index of $x$. As ...
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0answers
57 views

Does there exist a counterexample to the inequality $I(n) < 2{\left(\frac{n}{n + 1}\right)}^2$, if $n$ is odd and deficient?

If $n$ is odd and deficient, does there exist a counterexample to the inequality $$I(n) < 2{\left(\frac{n}{n + 1}\right)}^2,$$ where $$I(x) = \frac{\sigma(x)}{x}$$ is the abundancy index of ...
0
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1answer
38 views

A more elegant version of this function?

I challenged myself. The goal was to find a function $f$ with two variables $x$ and $y$ real, which results $1$ if $x=y$ and results $0$ if $x ≠ y$. But, the fonction can only use additions, ...
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0answers
45 views

What proportion of the positive integers satisfy $I(n^2) < (1 + \frac{1}{n})I(n)$, where $I(x)$ is the abundancy index of $x$?

Let $\sigma(x)$ denote the classical sum-of-divisors function, and let $$I(x) = \frac{\sigma(x)}{x}$$ be the abundancy index of the positive integer $x$. My question is this: What proportion of ...
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1answer
86 views

What proportion of the positive integers satisfy $I(n) < \frac{2n}{n + 1} \leq I(n^2)$ < 2?

Let $$I(x) = \frac{\sigma(x)}{x}$$ be the abundancy index of the positive integer $x$. Note that $\sigma(x)$ is the classical sum-of-divisors function. For example, $$\sigma(12) = 1 + 2 + 3 + 4 + ...
4
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1answer
102 views

Decidable & Recursive predicates

Let $C$ be a decidable predicate in the language of arithmetic HA, that is $$ \vdash (\forall \underline x)\: C(x) \vee \neg C(x).$$ $C$ is recursive if there exists a computable characteristic ...
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1answer
72 views

What proportion of the natural numbers satisfy the following inequalities?

Let $\sigma_1(n)$ be the sum of the divisors of $n \in \mathbb{N}$, and let $$I(n) = \frac{\sigma_1(n)}{n}$$ be the abundancy index of $n$. What proportion of the natural numbers satisfy the ...
4
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1answer
121 views

Image of the Euler phi function

I was reading a text about arithmetic functions, which ofcourse mentioned the Euler phi function. I was wondering whether $\phi(n)$ takes on all positive integer values. The answer doesn't seem so ...
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0answers
109 views

Can an odd perfect number be divisible by 101?

Preamble - This question is an offshoot from the following earlier questions here at MSE: Can an odd perfect number be divisible by 825? Can an odd perfect number be divisible by 165? Odd perfect ...
4
votes
2answers
70 views

An uncountable well-ordered subordering of asymptotic growth rates?

Define the following relation $\le$ between arithmetic functions $f$ and $g$ (mappings from $\mathbb{N} \rightarrow \mathbb{N}$): $f \le g := \exists n_0, k: \forall n: n \gt n_0 \implies f(n) \lt k ...
1
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1answer
103 views

Attempted exercise using Littlewood's theorem

This was an exercise to try to show we can use Littlewood's theorem$^1$ to prove that $$\lim_{N \to \infty}\frac{1}{N}\sum_{n=1}^{N} \frac{g(n)}{\log p_n} = 1 \hspace{30mm}(1)$$ If $\vartheta(p_k) ...
2
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1answer
102 views

Definition of “totient”

I had always taken the term "totient" to be defined by saying that the totient of a positive integer $n$ is the number of positive integers less than $n$ that are coprime to $n$. Thus, for example, ...