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

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2answers
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Offset a range of numbers

I want to find an equation to offset a range of numbers by a given amount. I'm not sure if I am using the term offset correctly. Lets say the range is from 0 - 1 and I want it to be offset by .25 : 0 ...
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1answer
57 views

Prime Factorization and Number Theory

Prime factorization of $n$ is $$n = p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$ Let $f(n) = p_1^{e_1}p_2^{e_2}p_3^{e_3}\cdots p_k^{e_k}$ where $e_k=a_k$ if $p_k|a_k$, else $e_k=a_k-1$ I want to ...
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1answer
38 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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1answer
47 views

Product of all divisors

Prove that $\prod_{d \mid n}d=n^{v(n)/2}$ where $v(n)$ is the sum of divisors function. We have if $n=p_{1}^{a_{1}}p_{2}^{a_{2}} \dots p_{k}^{a_{k}}$ then $v(n)=(a_{1} +1)(a_{2}+1) \dots (a_{k} ...
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0answers
125 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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0answers
135 views

Is the application of $\mu$ on $P_x(s)^k$ analogous to the differentiation $\frac{d^k f(\lambda) }{d\lambda^k}\biggr|_{\lambda=0}$?

Let me start with the following on elementary symmetric polynomials: The elementary symmetric polynomials appear when we expand a linear factorization of a monic polynomial: we have the identity ...
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0answers
293 views

Numbers $n$ such that Mertens' function is zero.

OEIS (A028442) lists the Numbers n such that Mertens' function $$ M(n)=\sum_{k=1}^n\mu(k) $$ is zero: 2, 39, 40, 58, 65, 93, 101, 145, 149, 150, 159, 160, 163, 164, 166, 214, 231, 232, 235, 236, ...
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0answers
71 views

On the indivisibility of odd perfect numbers by small numbers

A good day to everyone! This question is an offshoot of the following MSE posts: Odd perfect number divisors Can an odd perfect number be divisible by $101$? My question is as follows: Is ...
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0answers
103 views

Does the Fourier matrix $F_n$ represent a (tensor) multiplicative function?

At "Complex Hadamard Matrices", I found that, two Kronecker (tensor) products of Fourier matrices $k_1$ and $k_2$ are equivalent, if and only if $k_2$ can be obtained from $k_1$ by a combination of an ...
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0answers
78 views

Smallest prime p with N a quadratic residue mod p

Let $N$ be a square-free number equal to 2 or 3 $(\mod 4)$. Let $P(N)$ be the first odd prime, not a factor of $N$, for which $N$ is a quadratic residue. On average, $N$ would be a non-residue for ...
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0answers
69 views

Approximate how the Numbers $n$ such that Mertens' function is zero grow.

Is it possible to approximate how the "Numbers $n$ such that Mertens' function is zero" grow?
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0answers
22 views

gcd of product of exponents of prime factors and product of prime factors

Let $n = \prod\limits_i p_i^{k_i}$. I want to express $$ \gcd(\prod\limits_i k_i, \prod\limits_i p_i) $$ as an arithmetic function (i.e get rid of gcd). Is that possible? Thanks!
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0answers
56 views

A challenge question in elementary number theory!

Find an expression for the following sum: $$\sum_{i:(i,n)=1}(i-1,n)$$ I guess that this sum equals to $\phi(n)d(n).$
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0answers
54 views

sum over primes involving divisor function (variation of the Titchmarsh divisor problem)

Does there exist an asymptotic estimate for the following sum over primes $$ \sum_{p\leq x} \frac{\tau(p-1)}{p}\;, $$ where $\tau(n)=\sum_{d|n}1$ is the divisor function?
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0answers
28 views

All all hypo-multiplicative functions linear combinations of quasi-multiplicative functions?

A function $f(n)$ is multiplicative if, for all coprime $m$ and $n$, $f(m)f(n)=f(mn).$ It is called quasi-multiplicative by many authors if $f(m)f(n)=f(1)f(mn)$, and the two coincide when $f(1)=1.$ ...
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0answers
42 views

Multiple Dirichlet convolutions

I have been playing around with Dirichlet convolutions. As a reminder, take two arithmetic functions $f,g$, then their Dirichlet convolution is defined as the arithmetic function with: $(f\star g)(n) ...
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0answers
59 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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0answers
115 views

Generalizing a result on sums involving Euler's function

Motivation: It's known that there is a constant $0<K$ such that for any natural number $N$, $KN\leq \frac{\varphi(1)}{1}+\frac{\varphi(2)}{2}+\cdots+\frac{\varphi(N)}{N}$ (with $\varphi$ being ...
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0answers
66 views

Proving an identity involving the product of the Möbius function and Euler’s totient function.

Could anyone kindly help me to prove that $$ \sum_{d|n} \mu(d) \varphi(d) = 0 $$ for all even integers $ n \geq 2 $, where $ \mu $ is the Möbius function and $ \varphi $ is Euler’s totient function? ...
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0answers
17 views

Number of triples $(a, b, c)$ with $1 \leq a,b,c \leq n$ which are coprime ($gcd(a,b,c)=1$)

Number of ordered triples $(a, b, c)$ with $gcd(a, b, c) = 1$ and $1 \leq a, b, c \leq n$ can be computed using the following formula: $$ C(n) = \sum_{k=1}^n\mu(k) \left \lfloor \frac{n}{k} \right ...
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0answers
43 views

What is the inverse of $f(x)=x^{x^x}$?

I'm curious to find the inverse of $ f(x)=x^{x^x} $ As an added extra, I'm already familiar with the Lambert Product Log function.
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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 ...
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0answers
58 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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0answers
141 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 ...