All Questions
16
questions
1
vote
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
vote
1
answer
87
views
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
vote
0
answers
119
views
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
votes
0
answers
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
4
votes
0
answers
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
votes
0
answers
92
views
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 ...
2
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1
answer
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
votes
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
votes
2
answers
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
3
votes
1
answer
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
0
votes
1
answer
163
views
On prime-perfect numbers and the equation $\frac{\varphi(n)}{n}=\frac{\varphi(\operatorname{rad}(n))}{\operatorname{rad}(\sigma(n))}$
While I was exploring equations involving multiple compositions of number theoretic functions that satisfy the sequence of even perfect numbers, I wondered next question (below in the Appendix I add a ...
user243301
3
votes
1
answer
106
views
A problem similar than that of the amicable pairs using the function $\operatorname{rad}(k)$: a first statement or conjecture
In this post we denote the product of distinct primes dividing an integer $k> 1$ as $\operatorname{rad}(k)$, with the definition $\operatorname{rad}(1)=1$, that is the so-called radical of an ...
user243301
1
vote
1
answer
91
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On miscellaneous questions about perfect numbers III
This is a wild guess about odd perfect numbers. Thus you can see it as an exercise and not as a serious conjecture. I add here the MathWorld's reference dedicated to odd perfect numbers.
Question. ...
user243301
2
votes
1
answer
68
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On miscellaneous questions about perfect numbers II
Let $\varphi(m)$ the Euler's totient function and $\sigma(m)$ the sum of divisors function. We also denote the product of primes dividing an integer $m>1$ as $\operatorname{rad}(m)$, that is the ...
user243301
2
votes
0
answers
260
views
What about the equation $\sigma(2n)=2\left(n+\sigma(n)\right),$ involving the sum of divisor function?
Yesterday I wrote this equation involving the sum of divisors function $\sigma(l)=\sum_{d\mid l}d$,
$$\sigma(2n)=2\left(n+\sigma(n)\right).\tag{1}$$
Due to its very simple form I don't know if it is ...
user243301