# Questions tagged [divisor-counting-function]

For questions that involve the divisor counting function, also known as $\sigma_0$, $\tau$, or $d$.

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### estimating an elementary sum involving divisor function

Please guide me as to how to obtain the below bound and whether it is optimal. Let a squarefree integer $N=\prod_{1 \leq i \leq m} p_i$ be a product of $m$ primes ($p_1 < p_2 < \dots < p_m$) ...
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### Show that there are only finitely many pairs of positive integers $(n, m)$ such that $d(m!) = n!$.

Show that there are only finitely many pairs of positive integers $(n, m)$ such that $d(m!) = n!$, where $d(n)$ denotes the number of positive divisors of $n$. My approach (it isn’t complete and ...
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### How many divisors does $66^{49$} have? [duplicate]

The question was What is the amount of natural divisors of $66^{49}$ I know that I most likely have to calculate the prime divisors of $66$ (so $2$, $3$ and $11$). However I do not know how to go ...
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### Highly composite numbers which are the middle of a twin prime

From the first $10\ 000$ highly composite numbers listed in OEIS , the following $20$ are the middle of a twin-prime that is we have a highly composite number $N$ such that both $N-1$ and $N+1$ are ...
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### Stepwise irregularity of the divisor function, or does $\limsup_{n\to\infty} |d(n+1)-d(n)| = \infty$?

Defining $d(n):= \sum_{d|n} 1$, I know that $\liminf_{n\to\infty} d(n) = 2$, and Wigert showed that $\limsup_{n\to\infty} \frac{\log(d(n))}{ \log n/\log\log n} = \log 2$ (natural logarithm). This ...
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### Question involving series of divisor function and Euler function

We shall prove that $\sum_{n=1}^{+\infty} \frac{d(n)}{2^n}=\sum_{n=1}^{+\infty} \frac{1}{\phi(2^{n+1}-1)}$, where d(n) the divisor function. I was thinking of making use of the fact that d(n) is ...
1 vote
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### Count the Number of Positive Integer Solutions to $\prod_{i=1}^{k}x_i\le N$

Given a fixed integer $k$ and a large positive integer $N$, I'd like to count the number of positive integer solutions to the equation $\prod_{i=1}^{k}x_i\le N$ in sublinear time. Note that when $k=2$,...
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### Count the number of integer solutions to $x_1+x_2+x_3+x_4+x_5+x_6=25$ when, $x_1,x_2,x_3$ are odd and $x_4,x_5,x_6$ are even and $x_i is N$

How to count the number of integer solutions to $$x_1+x_2+x_3+x_4+x_5+x_6=25$$ When, $x_1,x_2,x_3$ are odd ($2k+1$) and $x_4,x_5,x_6$ are even ($2k$), $x_i \in N$. Is there a general formula to ...
1 vote