Question

About $\lim\dfrac{\sigma(n)}{n}$, where $\sigma(n)$ is the sum of divisor of $n$

score 83 · Answer 1 · 2021-01-29 10:36:49

83

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About $\lim\dfrac{\sigma(n)}{n}$, where $\sigma(n)$ is the sum of divisor of $n$

Published on 29 Jan 2021 - 10:36

score 195 · Answer 2 · 2021-01-30 14:10:30

195

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Number theory olympiad problem

Published on 30 Jan 2021 - 14:10

score 128 · Answer 3 · 2021-01-31 15:27:17

128

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Prove that exist finitely many positive integers $n$ satisfying $\tau (n)=a$ and $n|\phi (n)+\sigma (n)$

Published on 31 Jan 2021 - 15:27

score 32 · Answer 4 · 2021-02-08 00:32:57

Is there a similar relation to Robin's Inequality for the number of divisors ( $\sigma_0(n) $), instead of the sum of divisors (( $\sigma_1(n) $)

score 18 · Answer 5 · 2026-03-18 03:36:50

18

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How can the following summation be written in terms of $N$?

Published on 18 Mar 2026 - 3:36

score 111 · Answer 6 · 2021-02-14 10:03:50

111

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Is $n^2\mid \sigma(n)^{\sigma(n)}-1$ possible for $n>1$?

Published on 14 Feb 2021 - 10:03

score 87 · Answer 7 · 2026-03-27 13:10:49

87

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On the inequality $\frac{\sigma(n^2)}{q^k} < \frac{n^2 - q^k}{C}$ where $C>1$ and $q^k n^2$ is an odd perfect number - Part II

Published on 27 Mar 2026 - 13:10

score 554 · Answer 8 · 2026-03-27 13:10:49

554

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On $\frac{D(n^2)}{s(q)} \geq \frac{2n^2}{\sigma(q)} \geq \frac{\sigma(n^2)}{q} \geq \frac{2s(n^2)}{D(q)}$ where $q^k n^2$ is an odd perfect number

Published on 27 Mar 2026 - 13:10

score 316 · Answer 9 · 2021-02-22 16:30:55

316

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The average order of $\sigma(n)$.

Published on 22 Feb 2021 - 16:30

score 79 · Answer 10 · 2026-03-26 13:02:47

79

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Square of number of divisors of n equals n

Published on 26 Mar 2026 - 13:02

score 58 · Answer 11 · 2026-03-27 13:10:49

58

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If $k + 1$ is prime and $(k + 1) \mid (q - 1)$, then $\sigma(q^k)$ is divisible by $k + 1$, but not by $(k + 1)^2$ (unless $k+1=2$).

Published on 27 Mar 2026 - 13:10

score 233 · Answer 12 · 2021-03-26 21:39:24

233

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Abundant products of iterations of Euler's totient function

Published on 26 Mar 2021 - 21:39

score 50 · Answer 13 · 2021-04-07 14:43:51

50

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Help Understanding A Statement by D.H.J. POLYMATH

Published on 07 Apr 2021 - 14:43

score 72 · Answer 14 · 2026-03-27 13:11:26

72

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Is there an odd $x$ such that $2x^2 \equiv 0 \pmod {\sigma(x)}$ and $\sigma(x^2) \equiv 0 \pmod {\sigma(x) - 1}$?

Published on 27 Mar 2026 - 13:11

score 142 · Answer 15 · 2026-03-27 13:10:37

142

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On bounds for the quantity $n^2 / D(n^2)$ when $q^k n^2$ is an odd perfect number with special prime $q$ and $k > 1$

Published on 27 Mar 2026 - 13:10

List Question

About $\lim\dfrac{\sigma(n)}{n}$, where $\sigma(n)$ is the sum of divisor of $n$

Number theory olympiad problem

Prove that exist finitely many positive integers $n$ satisfying $\tau (n)=a$ and $n|\phi (n)+\sigma (n)$

Is there a similar relation to Robin's Inequality for the number of divisors ( $\sigma_0(n) $), instead of the sum of divisors (( $\sigma_1(n) $)

How can the following summation be written in terms of $N$?

Is $n^2\mid \sigma(n)^{\sigma(n)}-1$ possible for $n>1$?

On the inequality $\frac{\sigma(n^2)}{q^k} < \frac{n^2 - q^k}{C}$ where $C>1$ and $q^k n^2$ is an odd perfect number - Part II

On $\frac{D(n^2)}{s(q)} \geq \frac{2n^2}{\sigma(q)} \geq \frac{\sigma(n^2)}{q} \geq \frac{2s(n^2)}{D(q)}$ where $q^k n^2$ is an odd perfect number

The average order of $\sigma(n)$.

Square of number of divisors of n equals n

If $k + 1$ is prime and $(k + 1) \mid (q - 1)$, then $\sigma(q^k)$ is divisible by $k + 1$, but not by $(k + 1)^2$ (unless $k+1=2$).

Abundant products of iterations of Euler's totient function

Help Understanding A Statement by D.H.J. POLYMATH

Is there an odd $x$ such that $2x^2 \equiv 0 \pmod {\sigma(x)}$ and $\sigma(x^2) \equiv 0 \pmod {\sigma(x) - 1}$?

On bounds for the quantity $n^2 / D(n^2)$ when $q^k n^2$ is an odd perfect number with special prime $q$ and $k > 1$

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