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 ...
- 85.1k
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On the "efficiency" of digit-sum divisibility tricks in other bases (or about the growth rate of the number of divisors function).
Out of the different divisibility tricks there's a really simple rule that works for more than one divisor: The digit-sum. Specifically, if the sum of the digits of a number is a multiple of $1,3$ or $...
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Solutions of equations of form $\tau(n+a_i)=\tau(x+a_i)$ where $\tau$ is the number of divisors function and $a_i$ is a diverging series
Consider the function $$\tau(n)=\sum_{d|n}1$$ which gives the number of divisors of a number. The question is: How much information does $\tau(n)$ contain about $n$?
The answer is obviously: not very ...
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A Diophantine equation centered around the divisor counting function and squares
For each positive integer $n$, let $τ(n)$ be the prime counting function. Prove that for all positive integers $a$ and $b$ satisfy the following equation that $a+b$ is even:
$a + τ (a) = b^2 + 2$.
So ...
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Prove that φ(n) + d(n) ≤ n + 1 [duplicate]
Prove that φ(n) + d(n) ≤ n + 1.
d(n) is the number of positive divisors of n.
φ(n) is the Euler's Totient Function.
Attempt：
For a prime number n, φ(n) = n - 1 (all numbers less than n are relatively ...
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Solve $0=(u^2+v^2)^{\frac12}\left(u\sum_{n=1}^{\infty}\sigma_3(n)ny^n\cos(t-nz)+v\sum_{n=1}^{\infty}\sigma_3(n)ny^n\sin(t+nz)\right)$ for $t$
Let $t\in\left(\frac{\pi}3,\frac{2\pi}3\right)$. Let $y:=e^{-2\pi\sin t}$. Let $z:=2\pi\cos t$. Let $u:=1+240\sum_{n=1}^{\infty}\sigma_3(n)y^n\cos(nz)$. Let $v:=240\sum_{n=1}^{\infty}\sigma_3(n)y^n\...
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Dirichlet series of $\ln(n) \tau(n)$
I was experimenting with a technique I developed for double/multiple summation problems, and thought of this problem:
Find
$$S(p)=\sum_{n=1}^{\infty} \frac{\ln(n) \tau(n)}{n^p}$$
where $\tau(n)=\sum_{...
user1214193
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Numbers of divisors for a Mersenne number
Recently I encountered a problem:
If n is a positive integer, then is the number of divisors of $2^n - 1$ less or greater than the number of divisors of n?
I tried factoring and taking modulo n but ...
- 21
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Proving that there are infinite primes with digit sum 8 in base 10
I recently wrote about a problem I cam up with while thinking about number theory, which you can find on this post. Long story short, I'm trying to prove there are infinite natural numbers such that ...
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Call $n\in\Bbb N$ "balanced" if the sum of its digits equals the count of its divisors. How many "balanced" numbers are there up to $m$?
I recently stumbled across a problem about numbers' divisor count (more specifically, how many positive integers are equal to the square of their divisor count - answer was 2: they are 1 and 9).
But I ...
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Product of All Positive Divisors of a Number [duplicate]
Based on my current knowledge, the formula for finding the product of all positive divisors of a number $n$ is $= n^{\frac{\tau (n)}{2}}$ where the function $\tau (n)$ outputs the number of positive ...
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Asymptotic on $\sum_{n<x} 1/d_{\alpha}(n)$, $d_\alpha$ is the general divisor function.
Let $d(n)$ be the divisor function, that is, $$d(n)=\sum_{1\leq k\leq n, \,k|n} 1.$$ Or equivalently, $d(n)$ can be defined as the coefficients of $\zeta^2(s)$: $$\zeta^2(s)=\sum_{n\geq1} d(n)n^{-s}.$$...
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