Timeline for About inequalities that involve the sum of divisors, the Euler's totient and the aliquot part $\sigma(n)-n$
Current License: CC BY-SA 4.0
16 events
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| Mar 13, 2022 at 12:27 | comment | added | user142929 | (2/2) I don't know if these (what) equations or thoughts are in the literature/OEIS are related to $\sigma(a)\sigma(b)-\sigma(ab)=0$ for composite integers, for prime numbers we have $\sigma(x)\sigma(x)-\sigma(xx)=x$, and for the mentioned special case for prime numbers satisfying $\sigma(x^2)=y$ we've $\varphi(\sigma(xx))-\varphi(xx)=2x$. Can you think about a geometrical interpretation of this (how the sum of divisors function does carry rectangles, and the reletionship to chirality)? | |
| Mar 13, 2022 at 12:27 | comment | added | user142929 | (1/2) @mathlove I try to relate prime numbers $x>3$, imagine it as a letter L=1+\varphi(x), with $\varphi(x)$ the Euler's totient, and chirality. My thoughts are $x>1$ is a prime number if and only if $\sigma(x^2)-\sigma(x)^2=x$; also if $x,y$ are prime numbers such that $\sigma(x^2)=y$ then $\varphi(\sigma(x^2))-\varphi(x^2)=2x$ holds; finally $x=a\times b$ is a composite (non-chiral) integer with $1<a<b$ and $\gcd(a,b)=1$ then since the sum of divisors function is multiplicative we have $\sigma(a\times b)=\sigma(a)\times \sigma(b)$. Thus the equations that I was evoking | |
| S Dec 31, 2021 at 13:07 | history | bounty ended | CommunityBot | ||
| S Dec 31, 2021 at 13:07 | history | notice removed | CommunityBot | ||
| Dec 30, 2021 at 16:41 | history | edited | user142929 | CC BY-SA 4.0 |
I've edited a typo.
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| Dec 30, 2021 at 16:39 | comment | added | user142929 | Alternatively, an integer $n>1$ is an odd perfect number if and only if $$\sigma(n)=2\frac{\varphi(n)}{\varphi(2n)^2}\varphi\left(2n^2\frac{N_l}{N_k}\right)\frac{\varphi(N_k)}{\varphi(N_l)}$$ holds, for some choice of integers $1\leq k<l$ with $\gcd(n,\frac{N_l}{N_k})=1$ (take $\operatorname{rad}(n)\mid N_k$). | |
| Dec 25, 2021 at 12:59 | comment | added | user142929 | Many thanks @mathlove if you can fix it, in other case I do it in next few days, now I'm in a call center. | |
| Dec 25, 2021 at 12:58 | comment | added | user142929 | As aside comment I add a characterization for odd perfect numbers (similar than previous identitites): An integer $n>1$ is an odd perfect number if and only if $$\frac{1}{\varphi(n)}\cdot\sigma(n)=2\frac{\varphi(2n^2)}{\varphi(2n)^2}.$$ | |
| Dec 24, 2021 at 13:38 | comment | added | mathlove | It seems that you have a typo in Question A), "the condition $(3)$ and also the inequality $(3)$". | |
| S Dec 23, 2021 at 11:12 | history | bounty started | user142929 | ||
| S Dec 23, 2021 at 11:12 | history | notice added | user142929 | Authoritative reference needed | |
| Dec 23, 2021 at 11:12 | comment | added | user142929 | Many thanks for the attention and upvotes. If I can I'm going to post as a comment a characterization of odd perfect numbers similar than previous equations. It will be as aside comment of the post, now I'm going to offer a bounty. | |
| Nov 17, 2021 at 17:02 | history | edited | user142929 | CC BY-SA 4.0 |
I've added aside remarks about different, but similar, inequalities involving primorials.
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| Nov 1, 2021 at 14:39 | comment | added | user142929 | Many thanks for your edit. | |
| Nov 1, 2021 at 14:39 | history | edited | Martin Sleziak |
edited tags
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| Nov 1, 2021 at 14:34 | history | asked | user142929 | CC BY-SA 4.0 |