Questions tagged [arithmetic-functions]

For questions on arithmetic functions, i.e. real or complex valued functions defined on the set of natural numbers.

In number theory, an arithmetic, arithmetical, or number-theoretic function is a real or complex valued function $f(n)$ defined on the set of natural numbers (i.e. positive integers) that "expresses some arithmetical property of $n$."

To emphasize that they are being thought of as functions rather than sequences, values of an arithmetic function are usually denoted by $a(n)$ rather than $a_n$.

Some examples are Euler totient function, Jordan totient function, and Ramanujan tau function.

There is a larger class of number-theoretic functions that do not fit the above definition, e.g. the prime-counting functions.

557 questions
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Prime powers, patterns similar to $\lbrace 0,1,0,2,0,1,0,3\ldots \rbrace$ and formulas for $\sigma_k(n)$

Some time ago when decomponsing the natural numbers, $\mathbb{N}$, in prime powes I noticed a pattern in their powers. Taking, for example, the numbers $\lbrace 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 \rbrace$ and factorize them, we will get…
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Primes approximated by eigenvalues?

Consider the infinite matrix starting: $$\displaystyle T = -\left( \begin{array}{ccccccc} +1&+1&+1&+1&+1&+1&+1&\cdots \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&-2&+1&+1&-2&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&+1&+1&-4&+1&+1 \\ +1&-1&-2&-1&+1&+2&+1 \\…
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Sums of the form $\sum_{d|n} x^d$

Let $$S(x,n) = \sum_{d|n} x^d, \quad n \in \Bbb N. $$ Do these sums appear in the literature? What are they called if they do and what is known about them? To clarify, note that this sum is not the same as the generalized divisor function $$…
Asvin
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For all $n$ there exists $x$ such that $\varphi(x)<\varphi(x+1)<\ldots<\varphi(x+n)$

Let $\varphi$ be the Euler's function, i.e. $\varphi(n)$ stands for the number of integers $m \in \{1,\ldots,n\}$ such that $\text{gcd}(m,n)=1$. Let $n\ge 2$ be a positive integer. Show that there exists infinitely many $x$ for…
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Is there a "nice" formula for $\sum_{d|n}\mu(d)\phi(d)$?

I'm trying to work through Ireland and Rosen's A Classical Introduction to Modern Number Theory as I've heard good things about it. This is Exercise 12 from Chapter 2. Here $\mu$ is the Moebius function, and $\phi$ the totient function. Find…
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Polynomials whose fractional part behaves like a logarithm

For a given integer $m$, I'm looking for a classification of all polynomials $P$ with rational coefficients satisfying the logarithm-like condition $$P(ab)=P(a)+P(b) \pmod 1$$ for any integers $a, b$ coprime to $m$. I'm interested in these…
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How to prove $ \prod_{d|n} d= n^{\frac{\tau (n)}{2}}$

how to prove: $$ \prod_{d|n} d= n^{\frac{\tau (n)}{2}}$$ $\prod_{d|n} d$ is product of all of distinct positive divisor of $n$, $\tau (n)$ is number (count)of all of positive divisor of $n$
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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 updating the notes to fit better with the second edition…
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Proving that $\omega(N)\neq4$ for an odd perfect number $N$ by hand

Let $\omega(n)$ denote the number of distinct prime factors of a positive integer $n$, and let $N$ be an odd perfect number. It is not difficult to show that $\omega(N)\ge3$. In fact, Nocco already proved this in 1863. Showing that $\omega(N)\neq3$…
Librecoin
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Does the equation $\sigma(\sigma(x^2))=2x\sigma(x)$ have any odd solutions?

(Note: This question has been cross-posted to MO.) Let $\sigma(x)$ denote the classical sum of divisors of the positive integer $x$. Here is my question: Does the equation $\sigma(\sigma(x^2))=2x\sigma(x)$ have any odd solutions? MY ATTEMPT I…
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Intuitive basis of Mobius inversion?

If we're given $f(n)= \sum_{d|n}g\left(\frac{n}{d}\right),n \in \mathbb{N},$ then Mobius inversion gives $$g(n)=\sum_{d|n}\mu \left( d\right) f \left( \frac{n}{d}\right).$$ Also, the generalised Mobius inversion formula states that the above is…
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On the density of a particular subset of integers

Given a positive integer $n$ in the standard form $$n=\prod_k p_k^{\alpha_k}$$ and the arithmetic function $$f(n)=\sum_k \alpha_k p_k$$ let's define the subset $F$ of positive integers $$F=\Big\{n\in N:f(n)\,|\,n,\;f(n)\lt…
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Prove $\sum_{d \leq x} \mu(d)\left\lfloor \frac xd \right\rfloor = 1 $

I am trying to show $$\sum_{d \leq x} \mu(d)\left\lfloor \frac{x}{d} \right\rfloor = 1 \;\;\;\; \forall \; x \in \mathbb{R}, \; x \geq 1 $$ I know that the sum over the divisors $d$ of $n$ is zero if $n \neq 1$. So we can rule out integers that are…
Tyler Hilton
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A conjecture regarding odd perfect numbers

(Note: This question has now been cross-posted to MO.) Let $\sigma(z)$ denote the sum of the divisors of $z \in \mathbb{N}$, the set of positive integers. Denote the deficiency of $z$ by $D(z):=2z-\sigma(z)$, and the sum of the aliquot divisors of…
10
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Some regularity in the prime decomposition

This question is linked to my previous question, which is more specific to the sequence $(\rho(n))$. Definition of $\Phi$. Let's consider a function $\rho$, acting on the prime decomposition of an integer $n$: $$\begin{matrix} \rho\colon & \mathbb…
E. Joseph
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