Question

Bounds for the abundancy index of divisors of odd perfect numbers in terms of the deficiency function - Part II

score 107 · Answer 1 · 2026-03-27 21:20:30

107

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Bounds for the abundancy index of divisors of odd perfect numbers in terms of the deficiency function - Part II

Published on 27 Mar 2026 - 21:20

score 116 · Answer 2 · 2026-03-27 21:20:28

116

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Does the inequality $I(n^2) \leq 2 - \frac{5}{3q}$ improve $I(q^k) + I(n^2) < 3$, if $q^k n^2$ is an odd perfect number with special prime $q$?

Published on 27 Mar 2026 - 21:20

score 73 · Answer 3 · 2026-03-27 21:20:27

73

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On the reciprocal of $I(n^2) - \frac{2(q - 1)}{q}$, if $q^k n^2$ is a(n) (odd) perfect number with special prime $q$

Published on 27 Mar 2026 - 21:20

score 18 · Answer 4 · 2026-03-17 15:33:32

18

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sum of unbiased dice

Published on 17 Mar 2026 - 15:33

score 99 · Answer 5 · 2026-03-27 21:20:30

99

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Does $\frac{\sigma(q^k)}{n} + \frac{\sigma(n)}{q^k} < 10$ hold in general for an odd perfect number $q^k n^2$ with special prime $q$?

Published on 27 Mar 2026 - 21:20

score 126 · Answer 6 · 2026-03-29 13:50:19

126

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The map $m \to m\sigma(m)$ is not injective.

Published on 29 Mar 2026 - 13:50

score 40 · Answer 7 · 2026-03-28 10:24:09

40

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If $km$ is a Descartes number with quasi-Euler prime $m$, must $\sqrt{k}$ be a squarefree palindrome?

Published on 28 Mar 2026 - 10:24

score 48 · Answer 8 · 2026-03-27 21:20:28

48

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Please check my proof: If $q^k n^2$ is an odd perfect number with special prime $q$, then $\sigma(q^k)/2 \mid n$ if and only if $n \mid \sigma(n^2)$.

Published on 27 Mar 2026 - 21:20

score 56 · Answer 9 · 2026-03-27 21:20:29

56

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Does $G \mid I$ and $I \mid H$ still hold if $\sigma(q^k)/2$ is not squarefree, where $q^k n^2$ is an odd perfect number with special prime $q$?

Published on 27 Mar 2026 - 21:20

score 161 · Answer 10 · 2026-03-27 21:17:46

161

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On odd perfect numbers and a GCD - Part VII

Published on 27 Mar 2026 - 21:17

score 30 · Answer 11 · 2022-02-06 12:21:03

30

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R.T.P. σ(p^n)=(p^(n+1)-1)/(p-1). (where σ denotes the divisor function) For a prime p.

Published on 06 Feb 2022 - 12:21

score 115 · Answer 12 · 2026-02-23 04:57:24

115

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Does there exist a nontrivial prime power $q^k$ such that $\sigma(n^2)/n = q^k$ for some $n$?

Published on 23 Feb 2026 - 4:57

score 325 · Answer 13 · 2026-03-27 21:19:03

325

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What are the remaining cases to consider for this problem, specifically all the possible premises for $i(q)$?

Published on 27 Mar 2026 - 21:19

score 426 · Answer 14 · 2026-03-27 22:57:38

426

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Does $\sigma(m^2)/p^k$ divide $m^2 - p^k$, if $p^k m^2$ is an odd perfect number with special prime $p$?

Published on 27 Mar 2026 - 22:57

score 141 · Answer 15 · 2026-03-25 20:12:01

141

Views

Why is this alternating summation involving reciprocals of divisors always positive? A conjecture.

Published on 25 Mar 2026 - 20:12

List Question

Bounds for the abundancy index of divisors of odd perfect numbers in terms of the deficiency function - Part II

Does the inequality $I(n^2) \leq 2 - \frac{5}{3q}$ improve $I(q^k) + I(n^2) < 3$, if $q^k n^2$ is an odd perfect number with special prime $q$?

On the reciprocal of $I(n^2) - \frac{2(q - 1)}{q}$, if $q^k n^2$ is a(n) (odd) perfect number with special prime $q$

sum of unbiased dice

Does $\frac{\sigma(q^k)}{n} + \frac{\sigma(n)}{q^k} < 10$ hold in general for an odd perfect number $q^k n^2$ with special prime $q$?

The map $m \to m\sigma(m)$ is not injective.

If $km$ is a Descartes number with quasi-Euler prime $m$, must $\sqrt{k}$ be a squarefree palindrome?

Please check my proof: If $q^k n^2$ is an odd perfect number with special prime $q$, then $\sigma(q^k)/2 \mid n$ if and only if $n \mid \sigma(n^2)$.

Does $G \mid I$ and $I \mid H$ still hold if $\sigma(q^k)/2$ is not squarefree, where $q^k n^2$ is an odd perfect number with special prime $q$?

On odd perfect numbers and a GCD - Part VII

R.T.P. σ(p^n)=(p^(n+1)-1)/(p-1). (where σ denotes the divisor function) For a prime p.

Does there exist a nontrivial prime power $q^k$ such that $\sigma(n^2)/n = q^k$ for some $n$?

What are the remaining cases to consider for this problem, specifically all the possible premises for $i(q)$?

Does $\sigma(m^2)/p^k$ divide $m^2 - p^k$, if $p^k m^2$ is an odd perfect number with special prime $p$?

Why is this alternating summation involving reciprocals of divisors always positive? A conjecture.

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