All Questions
Tagged with divisor-sum divisor-counting-function
44
questions
3
votes
0
answers
32
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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 ($p_1 < p_2 < \dots < p_m$) ...
- 417
2
votes
1
answer
62
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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 $...
- 7,273
2
votes
1
answer
48
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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 ...
- 103
4
votes
2
answers
777
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How did Ramanujan came up with this?
The following is a picture of equation from Ramanujan's lost notebook. In this page, Ramanujan gives a closed form for,
$$\sum_{n\geq 1}\sigma_{s}(n)x^{n}$$
In an attempt initially it's claimed that,
...
- 1,521
0
votes
1
answer
339
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Finding a formula for $\sum_{d|n} \tau(d)$, where $\tau(d)$ is the number of divisors of $d$.
I am currently in the middle of the following exercise:
Exercise. Compute $$ \sum_{d|n} \left(\sigma(d)\mu\left(\frac{n}{d}\right)+\tau(d)\right),$$
where $\sigma$ is the function that corresponds to ...
- 1,141
1
vote
1
answer
70
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Hand calculation of divisor summatory function
Maybe this question is stupid, but there was a problem in a math competition (not even in the highest stage) in my county which asked to
Find $ \sum_{n\leq390} d(n)$, where $d(n)$ is the number of ...
- 45
1
vote
0
answers
53
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How can we show this relationship between the recursive sum of divisors function and Figurate Number Polynomial on Primes?
Let us define the following recursive function involving the sum of divisors function $\sigma(n)$:
\begin{array}{ l }
r(n,1)=\sigma(n) \\
r(n,2)=\sum_{d|n}r(d,1) \\
r(n,3)=\sum_{d|n}r(d,2) \\
\...
- 1,594
1
vote
1
answer
66
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Why does this identity with the product $\prod_{p\mid n}(p+k)$ and recursive sum of divisors function is true?
Let us define the following recursive function involving the sum of divisors function $\sigma(n)$:
\begin{array}{ l }
r(n,1)=\sigma(n) \\
r(n,2)=\sum_{d|n}r(d,1) \\
r(n,3)=\sum_{d|n}r(d,2) \\
\...
- 1,594
5
votes
1
answer
110
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How can we show this relationship between the sum of divisors function and the sum $p^{m}+2p^{m-1}+3p^{m-2}+\ldots+(m+1)$?
The sum of divisors function is commonly denoted by $\sigma(n)$. Now let us introduce a recursive definition of divisor functions:
$r_{n,1}=\sigma(n)$
$r_{n,2}=\sum_{d|n}r_{d,1}$
$r_{n,3}=\sum_{d|n}...
- 1,594
2
votes
0
answers
70
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Can this divisor and floor function be calculated faster
Let $d(n)$ is the number of divisors of $n$. Example: $d(6) = 4$ because $6$ has $4$ divisors {$1, 2, 3, 6$} (Sequence A000005).
Let $D(n)$ is the sum of first $n$ functions $d(x)$. Or we have $D(n) = ...
- 85
0
votes
1
answer
240
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If we know about the divisors of $n$, what can we comment about the divisors of $n+x$?
I was dealing with a different problem when this question struck me-
Let us have a natural number $n$, and we know that the divisors of $n$
are $d_1,d_2,\dots d_k$. Now, we add any other $x\in \...
- 9,592
2
votes
0
answers
123
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Average Number of Small Divisors
I'm working on a pet project of mine and I've come across a seemingly simple problem that I can neither solve nor find any reference to in the literature. The problem is this: Given $x$ sufficiently ...
- 953
1
vote
1
answer
87
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Square of number of divisors of n equals n
Find $n \in \mathbb{N}$ such that $n = \tau(n)^2$ ($\tau(n)$ being the number of positive divisors of $n$).
I tried some values for $n$, it seems that besides $n = 1$ and $n = 9$ there's no other ...
- 99
1
vote
1
answer
57
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When is the number of Divisors of a Number equivalent to one of its Factors?
My math teacher asked me this problem for homework and I am unsure how to solve it.
Which numbers contain a number of factors equivalent to the value of one of their divisors?
I found that 8 works, ...
- 171
0
votes
0
answers
130
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Limit superior of the number-of-divisors function $d(n)$
Let $d(n):=\sum _{d|n} 1$ be the number-of-divisors function. Wikipedia (here) defines the limit superior of $d(n)$ by
$$\underset{n\to \infty }{\overline{\text{lim}}}\frac{\log (d(n))}{\log (n)/\log (...
