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1answer
143 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 ...
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1answer
124 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 ...
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0answers
33 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 ...
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1answer
50 views

Simple equation to distribute points in a game [closed]

I need to create a equation to distribute points for users in following game: There are x users that play a game. If only one of them hit he gets max points. If all of them hit each gets min points. ...
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0answers
203 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 ...
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0answers
248 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 ...
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1answer
163 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 > ...
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2answers
169 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 ...
3
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1answer
273 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
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1answer
229 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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0answers
169 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
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1answer
547 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 ...
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5answers
1k 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
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3answers
531 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
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2answers
963 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 ...
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0answers
175 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
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2answers
409 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 ...
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3answers
587 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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2answers
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 ...
1
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2answers
700 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 ...