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4
votes
0answers
167 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
0answers
227 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 ...
0
votes
1answer
138 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 > ...
2
votes
2answers
152 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
votes
1answer
211 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
1answer
217 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,
$$\sigma(12) = 1 + 2 + 3 + 4 + ...
4
votes
0answers
157 views
Maximal order of Hooley's Delta function?
There is a large literature on Hooley's
$$
\Delta(n)=\max_u\sum_{d|n,\ e^u\le d< e^{u+1}}1
$$
giving its normal and average order. What is known of its maximal order?
Clearly $\Delta(n)\le d(n)$ ...
2
votes
1answer
525 views
A formula combining Euler $\phi$ and $\gcd$
Let us fix a natural number $N>1$ and $a_1, \ldots, a_n$ natural numbers satisfying $0 \leq a_i < N$, with the property that $1+ \sum a_i$ is divisible by $N$. Let $\phi$ be the Euler totient ...
1
vote
5answers
1k views
The Inverse of the Euler Totient Function
How can we calculate the cardinality of the inverse of Totient function of any positive integer n ?
I tried going through this paper, but I couldn't understand the procedure.
Thanks
7
votes
3answers
519 views
The digit sum: $s(na)=s(nb)$
Not that I was serious about the following question, but I think it is a must-to-ask as a follow-up to this MO post.
For integer $n\ge0$, let $s(n)$ denote the sum of the digits in the decimal ...
15
votes
2answers
917 views
Sum of $\sum_{k=1}^nd(k^2)$
There is a literature dealing with
$$
\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 ...
1
vote
0answers
167 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
2answers
390 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 ...
16
votes
3answers
570 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
votes
2answers
1k 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
vote
2answers
651 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 ...
