Questions tagged [divisor-sum]
For questions on the divisor sum function and its generalizations.
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Divisors Sum Related Interesting Approximate Relation
Working on Divisors Sum Efficient calulcation topic. Accidentaly discovered one interesting relation which is accurate up to $10^{17}$ order.
$$\sum_{i=1}^{\infty}{\frac{\sigma(i)}{e^{i}}}\approx\frac{...
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patterns in the abundancy index of integers
Let $\sigma(n)$ be the sum of all divisors (including 1 and $n$) of $n$, and define the abundancy index of $n$ as $I(n) = \sigma(n)/n $. For example: $I(6)= \frac{1+2+3+6}{6} = 1/1+1/2 +1/3 +1/6 = 2$. ...
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Proof for the sum and number of positive divisors for a positive integer $n$. [duplicate]
I know that the number of positive divisors of $n$ can be given by :
$\tau(n)$ = $(a_1+1)(a_2+1)\ldots(a_k+1)$ where $n = p_1^{a_1}p_2^{a_2}.... p_k^{a_k}$, where $p_1, p_2... p_k$ are the prime ...
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On the "efficiency" of digit-sum divisibility tricks in other bases (or about the growth rate of the number of divisors function).
Out of the different divisibility tricks there's a really simple rule that works for more than one divisor: The digit-sum. Specifically, if the sum of the digits of a number is a multiple of $1,3$ or $...
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Upper bounds on the greatest common divisor of sums of geometric series
Let $S_1=\sum_{i=0}^{n} p^i = \frac{p^{n+1}-1}{p-1}$ and $S_2=\sum_{i=0}^{m} q^i = \frac{q^{n+1}-1}{q-1}$ be two sums of geometric series, and $\gcd\left(S_1,S_2\right)$ its greatest common divisor. ...
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Solutions of equations of form $\tau(n+a_i)=\tau(x+a_i)$ where $\tau$ is the number of divisors function and $a_i$ is a diverging series
Consider the function $$\tau(n)=\sum_{d|n}1$$ which gives the number of divisors of a number. The question is: How much information does $\tau(n)$ contain about $n$?
The answer is obviously: not very ...
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The number of positive divisors of a number that are not present in another number.
How many positive divisors are there of $30^{2024}$ which are not divisors of $20^{2021}$?
I have tried many ways to try to get a pattern for this problem but I can't. I know that $30$ has $8$ ...
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Maxima of the Sum of Divisors Function and Upper Bound of a similar Ratio
If we define a function (aka Gronwall’s function) as: $$F(n)=\frac{\sigma(n)}{n \log \log n}$$
Then for $n>15$, it does have an upper bound. I want to know what's that specific upper bound is?
Also ...
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Growth rate of sum of divisors cubed [closed]
I a trying to find a result similar to:
$$\limsup_{n \to \infty} \frac{\sigma_1(n)}{n \log \log (n)} = e^\gamma$$
(where $\sigma_1$ is the sum of divisors function) but regarding the growth rate of $\...
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Why must $n-\lfloor\frac n2\rfloor+\lfloor\frac n3\rfloor-\dotsb$ grow like $n\ln2$? [duplicate]
Let
$$a(n):=n-\left\lfloor\frac n2\right\rfloor+\left\lfloor\frac n3\right\rfloor-\left\lfloor\frac n4\right\rfloor+\left\lfloor\frac n5\right\rfloor-\dotsb.$$
Note that this is a finite sum. Naïvely,
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Determine all positive integers $n$ such that: $n+d(n)+d(d(n))+\dotsb=2023$.
For a positive integer number $n>1$, we say that $d(n)$ is its superdivisor if $d(n)$ is the largest divisor of $n$ such that $d(n)<n$. Additionally, we define $d(0)=d(1)=0$. Determine all ...
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Square of prime numbers
This conjecture is inspired by the comment of @Eric Snyder: Prime numbers which end with 03, 23, 43, 63 or 83
$n$ is a natural number $>1$, $\varphi(n)$ denotes the Euler's totient function, $P_n$ ...
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Prime numbers which end with 03, 23, 43, 63 or 83
This is inspired from this post: Prime numbers which end with $19, 39, 59, 79$ or $99$
Here I found a new formula:
$n$ is a natural number $>1$, $\varphi(n)$ denotes the Euler's totient function, $...
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Can the sum of odd divisors of an integer exceed n [closed]
I know that the sum of divisors of an integer n can exceed it (abundant numbers) but can this occur when only considering odd divisors of n? Can the sum of all integers j : j|n, 2∤j be greater than n? ...
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Prime numbers which end with $59$ or $79$ [closed]
This is the weak conjecture of https://math.stackexchange.com/questions/4834936/prime-numbers-which-end-with-19-39-59-79-or-99.\
$\varphi(n)$ denotes the Euler’s totient function, $n$ denotes a ...
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