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**7**

votes

**1**answer

386 views

### Does 53 diverge to infinity in this Collatz-like sequence?

This function has been explored a bit at MSE (over the past week):
\begin{eqnarray}
f(n) &=& (n-1)^2 \; \textrm{if} \; (n \bmod 4) = 1\\
f(n) &=& \lfloor n/4 \rfloor \; \textrm{...

**4**

votes

**1**answer

243 views

### Does anyone recognize this exponential sum?

For $a$, $b$ two integers, let $(a,b)$ denotes their gcd. We define the following exponential sum :
$$G_q(n):=\sum_{d|q,~(d,q/d)=1}{e^{2i\pi n\frac{dd'}{q}}}$$
for $n$ a non-negative integer and $q$ ...

**4**

votes

**0**answers

159 views

### The properties of Pos

Given $n\in\mathbb{N}$, and $f:\mathbb{N}^*\rightarrow \mathbb{N}$, let define $Pos$ as:
$$Pos(f)(n)= |\{x \leq n, f(x)=f(n)\}|$$
When given $n\in\mathbb{N}$, this function gives the 'position' of $...

**1**

vote

**0**answers

177 views

### On even almost perfect numbers other than powers of two

(Note: This question is an improved version of and has been cross-posted from this MSE post.)
Let $\sigma(x)$ denote the sum of the divisors of $x$. If $\sigma(x) = 2x - 1$, then we call $x$ an ...

**7**

votes

**2**answers

403 views

### Asymptotics of product of Euler's totient function (A001088)?

Conjecture:
\begin{align}
\lim_{n\to \infty } \, \frac{\left(\prod _{k=1}^n \phi (k)\right){}^{1/n}}{n}\sim 0.2059\text{...}
\end{align}
The numerical result from 100000 terms is:
My questions ...

**4**

votes

**2**answers

367 views

### Are there multiplicative functions which are not rational?

Vaidyanathaswamy calls an arithmetic function rational if it is the convolution of some finite collection of functions which are either completely multiplicative or inverse to a completely ...

**0**

votes

**1**answer

116 views

### Sum of digits of a power [closed]

Are there any explicit formula for a sum of digits for a power in the given base? A problem to be specific: find a sum of digits for a number $2^{100}$ in the system with a base 5. In the system with ...

**6**

votes

**1**answer

226 views

### A question about $(0,1]$-valued multiplicative functions

Suppose $f:\mathbb{N}\to [0,1]$ is a multiplicative function (i.e. $f(nm)=f(n)f(m)$ whenever $m$ and $n$ are coprime). Suppose $f$ has non-zero mean, which means
$$
\lim_{N\to\infty}\frac{1}{N} \sum_{...

**13**

votes

**1**answer

633 views

### Is the set of multiplicatively even numbers thick?

A positive integer is multiplicatively even (odd) if, when decomposed into primes, the sum of the exponents is even (odd).
A subset of the integers is thick if it contains arbitrarily long intervals $\...

**2**

votes

**0**answers

63 views

### Discrete “difference” equations that involve changes in both shift and scale

A standard use of the Z-transform ($F(z) = \sum_n (f[n] \cdot z^{-n} )$) is to understand the effect of a difference equation on a signal. For instance:
$y[n] = x[n] + y[n-1]$
$Y(z) = X(z) + Y(z) \...

**1**

vote

**1**answer

137 views

### estimate an sum

I need estimate the following sum:
$\sum_{d=1}^{n}\frac{\mu(d)}{d}\sum_{k=1}^{\lfloor n/d\rfloor}\frac{1}{k}\frac{q^k}{1-q^{-kd}}$, where $q>1$ and $\mu$ is the Möbius function.
To obtain the ...

**2**

votes

**0**answers

178 views

### Infinite sums with Mobius Inversion : can we have uniform convergence of inversion formula?

My question is on Mobius inversion formula convergence/properties when used with infinite sums of function.
Lets consider (on $\mathbb{R}^{+}$):
$$S(x)= \sum\limits_{n=1}^{\infty} f(nx)$$
We call $...

**1**

vote

**1**answer

156 views

### Existence of arithmetic function satisfying a certain property

I was interested in an arithmetic function satisfying a certain property, I am not sure at the moment if such thing even exists or not. But I was wondering maybe I could get some hint or idea or input ...

**2**

votes

**1**answer

189 views

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

This question was also asked on MSE.
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 ...

**1**

vote

**0**answers

47 views

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

A function is called quasi-multiplicative by many authors if $f(m)f(n)=f(1)f(mn)$, a slight generalization of multiplicativity. (Basically a multiplicative function times a constant is a quasi-...

**4**

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**0**answers

232 views

### A closed formula for this arithmetic function

The following function comes up in my research as part of a sufficient condition for capability of $p$-group of class two and prime exponent. Given a nonnegative integer $m$, express $m$ as a ...

**5**

votes

**0**answers

277 views

### $n\varphi(n)\equiv 2\pmod{\sigma(n)}$ as a primality test

It is known from Subbarao, "On two congruences for primality" that $n>22$ is a prime iff $$n\sigma(n)\equiv 2\pmod{\varphi(n)},$$ where $\varphi(n)$ is Euler's function and $\sigma(n)$ is sum of ...

**-1**

votes

**1**answer

206 views

### 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 > ...

**1**

vote

**2**answers

181 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 ...

**4**

votes

**1**answer

428 views

### Menon’s identity

I also put this question in stackexchange, but remained unanswered. http://math.stackexchange.com/questions/506996/menons-identity
Let $G$ be a group of order $n$. Consider an action of $U_n$, the ...

**2**

votes

**1**answer

259 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**

votes

**0**answers

200 views

### 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 $\Delta(n)\le d(n)$ ...

**2**

votes

**1**answer

641 views

### A formula combining Euler $\phi$ and $\gcd$

Let us fix a natural number $N>1$ and $a_1, \ldots, a_n$ natural numbers satisfying $0 \leq a_i < N$, with the property that $1+ \sum a_i$ is divisible by $N$. Let $\phi$ be the Euler totient ...

**1**

vote

**5**answers

2k views

### The Inverse of the Euler Totient Function

How can we calculate the cardinality of the inverse of Totient function of any positive integer n ?
I tried going through this paper, but I couldn't understand the procedure.
Thanks

**7**

votes

**3**answers

563 views

### The digit sum: $s(na)=s(nb)$

Not that I was serious about the following question, but I think it is a must-to-ask as a follow-up to this MO post.
For integer $n\ge0$, let $s(n)$ denote the sum of the digits in the decimal ...

**15**

votes

**2**answers

1k views

### Sum of $\sum_{k=1}^nd(k^2)$

There is a literature dealing with
$$
\sum_{k\le x}d(f(k))
$$
where $f$ is an irreducible polynomial and $d(n)$ is the number of divisors of $n$. Erdos 1952 shows that the sum $\asymp x\log x,$ which ...

**3**

votes

**0**answers

269 views

### Linear combination of multiplicative functions

Carlitz showed necessary and sufficient conditions for an arithmetic function to be a linear combination of two multiplicative functions. He mentions the possibility of generalizing to $k$ ...

**1**

vote

**2**answers

489 views

### Which rationals are sum-of-divisor function quotients

Consider the function $\sigma(n)/n$, where $\sigma$ is the usual sum-of-divisors function. I read somewhere that it is unknown what rational numbers are in fact values of this function (or at any ...

**17**

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**3**answers

639 views

### Extending arithmetic functions to groups

Thinking along the lines of Tom Leinster's fascinating recent question, I'm wondering more generally about how to extend questions about natural numbers to groups, with the cyclic groups representing ...

**12**

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**2**answers

2k views

### Generalized Euler phi function

Let $n$ be an integer, there is a well-known formula for $\varphi(n)$ where $\varphi$ is the Euler phi function. Essentially, $\varphi(n)$ gives the number of invertible elements in $\mathbb{Z}/n\...

**1**

vote

**2**answers

782 views

### Sum of digits iterated

Original version.
I believe that it is an elementary question, already discussed somewhere. But I just have no idea of how to start it properly. Take a positive integer $n=n_1$ and compute its sum of ...