All Questions
Tagged with divisor-sum prime-factorization
28
questions
1
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79
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Continued aliquot sums
What happens if one takes the aliquot sum of an integer and then repeats the process so that one takes the aliquot sum of all of those factors that were not reduced to the number 1 on the previous ...
1
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1
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87
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Solution of $\sigma(\sigma(m)+m)=2\sigma(m)$ with $\omega(m)>8$?
This question is related to this one.
$\sigma(n)$ is the divisor-sum function (the sum of the positive divisors of $n$) and $\omega(n)$ is the number of distinct prime factors of $n$.
The object is ...
- 85.1k
1
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0
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119
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Is $n=2$ the only even solution of $\sigma(\sigma(n)+n)=2\sigma(n)$?
Inspired by this
question.
For positive integer $m$ , let $\sigma(m)$ be the divisor-sum function.
Let $S$ be the set of positive integers $n$ satisfying $$\sigma(\sigma(n)+n)=2\sigma(n)$$ In the ...
- 85.1k
2
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0
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68
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Largest possible prime factor for given $k$?
Let $k$ be a positive integer.
What is the largest possible prime factor of a squarefree positive integer $\ n\ $ with $\ \omega(n)=k\ $ (That is, it has exactly $\ k\ $ prime factors) satisfying the ...
- 85.1k
1
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1
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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
4
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0
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114
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Conjecture on the sum of prime factors
$\text{Notations}$
Let $\pi(n)$ be the prime counting function.
Let denote $\alpha(n)$ the sum of the prime factors of $n$. (i.e. In other words, if $$n=p_1^{x_1}p_2^{x_2}...p_m^{x_m}$$ then $\alpha(n)...
user799688
4
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0
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92
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On the least number of factors $\sigma(q^{e_q})$ to get the least multiple of $\operatorname{rad}(n)$, on assumption that $n$ is an odd perfect number
I don't know if my question can be answered easily from your reasonings and knowledeges of the theory of odd perfect numbers. I wondered about it yesterday. Now this post is cross-posted on ...
1
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0
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PROOF: A Relationship Between A Natural Number and The Quantity of Its Divisors' Divisors
An arbitrary natural number $N \in \mathbb{N}$ has the divisors $d_1,d_2,...,d_s$ , where $N$ has $s$ divisors in total. If $n_i$ denotes the number of dividers to $d_i, i = 1, ..., s$, it can be ...
user736333
1
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2
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82
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Tight Bounds on the Sum and Difference of Divisors of RSA Challenge Numbers
The title says it all: given that $n$ of a given length $d$ is a RSA Challenge Number where $n=pq$, where $p$ and $q$ are two primes of length $d/2$. My question is, knowing how the RSA numbers are ...
- 788
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74
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Upper Bound and Lower Bound of the Sum of the Prime Divisors of a Odd Semiprime
Lets say we have $n$, an odd semiprime. $p$ and $q$ are odd primes, such that $pq=n$. What are the tightest upper and lower bounds of $p+q$ in terms of $n$ known right now? Right now, I have $2\sqrt n\...
- 788
2
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1
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257
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On prime factors of odd perfect numbers
Why is it so difficult to determine actual numerical values for prime factors of odd perfect numbers?
Recall that, if $N$ is an odd perfect number, then Euler proved that it takes the form $N = q^k n^...
- 8,010
2
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1
answer
819
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Finding the sum of the reciprocals of the positive divisors of a number
Let $d_1, d_2, \dots , d_k$ be all the positive factors of a positive integer $n$ including $1$ and $n$. If $d_1+d_2+ \dots+d_k=72$, then find the value of $\frac{1}{d_1}+\frac{1}{d_2}+\dots+\frac{1}{...
- 3,023
2
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2
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227
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On $\text{Lower bound}\leq \operatorname{rad}(n)$, where $n$ is an odd perfect number: reference request or what work can be done about it
For integers $n\geq 1$ we denote the square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$ that is the product of distinct primes dividing an integer $n>1$ ...
user243301
6
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0
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152
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Estimation of the number of solutions for the equation $\sigma(\varphi(n))=\sigma(\operatorname{rad}(n))$
For integers $n\geq 1$ in this post we denote the square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$ that is the product of distinct primes dividing an ...
user243301
3
votes
1
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77
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On the equation $\sigma(m)=105k$ over odd integers $m\geq 1$, and the deficiency of its solutions
In this post we denote the sum of positive divisors of a natural $n\geq 1$ as $$\sigma(n)=\sum_{d\mid n}d.$$
We consider integers $m\geq 1$ and $k\geq 1$ where $m\equiv 1\text{ mod }2$ (thus $m$ is ...
user243301