Question

If $N = q^k n^2$ is an odd perfect number with Euler prime $q$, and $k=1$, does it follow that $\frac{\sigma(n^2)}{n^2} \geq 2 - \frac{5}{3q}$?

score 84 · Answer 1 · 2026-03-26 07:35:22

84

Views

If $N = q^k n^2$ is an odd perfect number with Euler prime $q$, and $k=1$, does it follow that $\frac{\sigma(n^2)}{n^2} \geq 2 - \frac{5}{3q}$?

Published on 26 Mar 2026 - 7:35

score 71 · Answer 2 · 2016-07-26 11:30:30

71

Views

Can these bounds, for the deficiency $D(x)=2x-\sigma(x)$ of a deficient number $x>1$, be improved?

Published on 26 Jul 2016 - 11:30

score 74 · Answer 3 · 2026-03-28 06:31:02

74

Views

On $\sigma(n)=\sigma \left( \left \lfloor{e^{H_n-\gamma}}\right \rfloor \right) $, for integers $n\geq 1$

Published on 28 Mar 2026 - 6:31

score 108 · Answer 4 · 2026-03-28 06:31:01

108

Views

Can you provide us calculations for an upper bound of $\frac{15}{\pi^2}ne^{H_n}\log(H_n)-e^{H_{n^2}}\log(H_{n^2})$?

Published on 28 Mar 2026 - 6:31

score 143 · Answer 5 · 2026-03-25 17:42:14

143

Views

On $-\log (2\pi)\sigma_0(n)-\frac{1}{2}\sum_{1<d\mid n}\log\left(\frac{1+2d}{(d+1)^2}\right),$ where $\sigma_0(n)=\sum_{d\mid n}1$ and its twists

Published on 25 Mar 2026 - 17:42

score 65 · Answer 6 · 2016-08-16 15:07:22

65

Views

On Solvability of an Equation with the Sigma Function

Published on 16 Aug 2016 - 15:07

score 88 · Answer 7 · 2026-03-26 08:03:23

88

Views

What is a sharp upper bound for $\prod_{i=1}^{r}{\left({p_i}^{\alpha_i} - \sigma({p_i}^{\alpha_i - 1})\right)}?$

Published on 26 Mar 2026 - 8:03

score 42 · Answer 8 · 2026-03-26 08:02:32

42

Views

If $D(m)$ is the deficiency of the deficient number $m$, then what is $\lim_{m \rightarrow \infty}{\frac{D(m)}{m}}$?

Published on 26 Mar 2026 - 8:02

score 52 · Answer 9 · 2026-03-26 07:58:50

52

Views

For what numbers $x$ is $D(x) = 2x - \sigma_1(x)$ equal to $\varphi(x)$?

Published on 26 Mar 2026 - 7:58

score 79 · Answer 10 · 2026-03-26 07:39:15

79

Views

Is the following statement true if $N = q^k n^2$ is an odd perfect number given in Eulerian form?

Published on 26 Mar 2026 - 7:39

score 66 · Answer 11 · 2026-03-25 04:42:50

66

Views

On interesting definitions involving the sum of remainders function and sequences of integers with special abundancy

Published on 25 Mar 2026 - 4:42

score 66 · Answer 12 · 2026-03-27 00:02:40

66

Views

On questions concerning a modified sum of divisors function and a modified Euler's totient function

Published on 27 Mar 2026 - 0:02

score 180 · Answer 13 · 2026-03-26 04:50:29

180

Views

On the complex function $f(s)=\sum\limits_{n=1}^\infty\sigma(n)^{-s}$

Published on 26 Mar 2026 - 4:50

score 83 · Answer 14 · 2026-03-26 07:35:35

83

Views

On identities and congruences involving the harmonic mean of odd perfect numbers

Published on 26 Mar 2026 - 7:35

score 158 · Answer 15 · 2026-03-26 07:39:14

158

Views

On a conjectured relationship between the least prime factor and the Euler prime of an odd perfect number

Published on 26 Mar 2026 - 7:39

List Question

If $N = q^k n^2$ is an odd perfect number with Euler prime $q$, and $k=1$, does it follow that $\frac{\sigma(n^2)}{n^2} \geq 2 - \frac{5}{3q}$?

Can these bounds, for the deficiency $D(x)=2x-\sigma(x)$ of a deficient number $x>1$, be improved?

On $\sigma(n)=\sigma \left( \left \lfloor{e^{H_n-\gamma}}\right \rfloor \right) $, for integers $n\geq 1$

Can you provide us calculations for an upper bound of $\frac{15}{\pi^2}ne^{H_n}\log(H_n)-e^{H_{n^2}}\log(H_{n^2})$?

On $-\log (2\pi)\sigma_0(n)-\frac{1}{2}\sum_{1<d\mid n}\log\left(\frac{1+2d}{(d+1)^2}\right),$ where $\sigma_0(n)=\sum_{d\mid n}1$ and its twists

On Solvability of an Equation with the Sigma Function

What is a sharp upper bound for $\prod_{i=1}^{r}{\left({p_i}^{\alpha_i} - \sigma({p_i}^{\alpha_i - 1})\right)}?$

If $D(m)$ is the deficiency of the deficient number $m$, then what is $\lim_{m \rightarrow \infty}{\frac{D(m)}{m}}$?

For what numbers $x$ is $D(x) = 2x - \sigma_1(x)$ equal to $\varphi(x)$?

Is the following statement true if $N = q^k n^2$ is an odd perfect number given in Eulerian form?

On interesting definitions involving the sum of remainders function and sequences of integers with special abundancy

On questions concerning a modified sum of divisors function and a modified Euler's totient function

On the complex function $f(s)=\sum\limits_{n=1}^\infty\sigma(n)^{-s}$

On identities and congruences involving the harmonic mean of odd perfect numbers

On a conjectured relationship between the least prime factor and the Euler prime of an odd perfect number

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