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Results tagged with arithmetic-functions
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user 10365
An arithmetic function is one whose domain is the positive integers and whose range is a subset of the complex numbers. There are a number of important number-theoretic examples.
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A conjecture regarding odd perfect numbers
This answer is a direct response to Professor Pace Nielsen's suggestion to perform a "sanity check" and complements this other answer by providing a proof of the following biconditionals:
If $p^k m^2 …
0
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A conjecture regarding odd perfect numbers
Further to this recent corrigendum in NNTDM, which corrects an oversight in Some modular considerations regarding odd perfect numbers – Part II, we realized that we do in fact have the following bicon …
1
vote
Accepted
Help with R. Ryan's "A simpler dense proof regarding the abundancy Index."
Hint: Since $q \mid \sigma(2^m)$, then
$$q \leq 2^{m+1} - 1,$$
which implies that
$$\frac{1}{q} \geq \frac{1}{2^{m+1} - 1}.$$
Can you finish?
6
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0
answers
503
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Does the equation $\sigma(\sigma(x^2))=2x\sigma(x)$ have any odd solutions?
This question was posted in MSE in early August 2020. It did garner several upvotes, but did not receive any responses. I have therefore cross-posted it here, hoping that it gets answered.
Let $\sig …
2
votes
A conjecture regarding odd perfect numbers
The following assertion appears in Theorem 3.3 (page 7, equations (5) to (6)) of Odd multiperfect numbers by Shi-Chao Chen and Hao Luo:
Let $n=\pi^{\alpha} M^2$ be an odd $2$-perfect number, with $\p …
3
votes
4
answers
1k
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A conjecture regarding odd perfect numbers
(Note: I asked this question in MSE this June 2018 but did not receive any responses there. I have therefore cross-posted it here, hoping that it gets answered.)
Let $\sigma(z)$ denote the sum of t …
1
vote
0
answers
255
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On even almost perfect numbers other than powers of two
(Note: This question is an improved version of and has been cross-posted from this MSE post.)
Let $\sigma(x)$ denote the sum of the divisors of $x$. If $\sigma(x) = 2x - 1$, then we call $x$ an almo …
6
votes
1
answer
2k
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If $N = qn^2$ is an odd perfect number with $\gcd(q,n)=1$, is it possible to have $q + 1 = \...
The title says it all.
Question
If $N = qn^2$ is an odd perfect number with Euler prime $q$ and $\gcd(q,n)=1$, is it possible to have $q + 1 = \sigma(n)$?
Heuristic
From the Descartes spoof, with qu …
0
votes
1
answer
312
views
Are all known $k$-multiperfect numbers (for $k > 2$) not squarefree?
I asked the following question in MSE four ($4$) days ago, but so far nobody has posted an answer.
The gist of the question is as follows:
Are all known $k$-multiperfect numbers (for $k > 2$) …
1
vote
2
answers
199
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What proportion of the positive integers satisfy $I(n^2) < (1 + \frac{1}{n})I(n)$, where $I(...
Let $\sigma(x)$ denote the classical sum-of-divisors function, and let
$$I(x) = \frac{\sigma(x)}{x}$$
be the abundancy index of the positive integer $x$.
My question is this: What proportion of th …
2
votes
1
answer
301
views
What proportion of the positive integers satisfy $I(n) < \frac{2n}{n + 1} \leq I(n^2) < 2$?
Let
$$I(x) = \frac{\sigma(x)}{x}$$
be the abundancy index of the positive integer $x$. Note that $\sigma(x)$ is the classical sum-of-divisors function. For example,
$$\sigma(12) = 1 + 2 + 3 + 4 + …